problem solving involving geometric figures

Using Geometric Concepts and Properties to Solve Problems

Introduction.

Often, you will be asked to solve problems involving geometric relationships or other shapes. For real-world problems, those geometric relationships mostly involve measurable attributes, such as length, area, or volume.

Sometimes, those problems will involve the perimeter or circumference, or the area of a 2-dimensional figure.

For example, what is the distance around the track that is shown?  Or, what is the area of the portion of the field that is covered with grass?

You may also see problems that involve the volume or surface area of a 3-dimensional figure.  For example, what is the area of the roof of the building that is shown?

Another common type of geometric problem involves using proportional reasoning.

For example, an artist created a painting that needs to be reduced proportionally for the flyer advertising an art gallery opening. If the dimensions of the painting are reduced by a factor of 40%, what will be the dimensions of the image on the flyer?

In this resource, you will investigate ways to apply a problem-solving model to determine the solutions for geometric problems like these.

A basic problem solving model contains the following four steps:

Solving Problems Using Perimeter and Circumference

You may recall that the perimeter of an object is the distance around the edge of the object. If the object contains circles, then you may need to think about the circumference of a circle, which is the perimeter of the circle.

A tire on a passenger car has a diameter of 18 inches. When the tire has rotated 5 times, how far will the car have traveled?

Step 1 : Read, understand, and interpret the problem.

• What information is presented?
• What is the problem asking me to find?
• What information may be extra information that I do not need?

Step 2 : Make a plan.

• Draw a picture.
• Use a formula: Which formula do I need to use? (Hint: Look at your Mathematics Reference Materials)

Step 3 : Implement your plan.

• What formulas do I need?
• What information can I interpret from the diagram, table, or other given information?
• Solve the problem.

Step 4 : Evaluate your answer.

• Does my answer make sense?
• Did I answer the question that was asked?
• Are my units correct?

A cylindrical barrel with a diameter of 20 inches is used to hold fuel for a barbecue cook off. The chef rolls the barrel so that it completes 7 rotations. How many feet did the chef roll the barrel?

Solving Problems Using Area and Surface Area

You may also encounter real-world geometric problems that ask you to find the area of 2-dimensional figures or the surface area of 3-dimensional figures. The key to solving these problems is to look for ways to break the region into smaller figures of which you know how to find the area.

Mr. Elder wants to cover a wall in his kitchen with wallpaper. The wall is shown in the figure below.

If wallpaper costs $1.75 per square foot, how much will Mr. Elder spend on wallpaper to completely cover this wall, excluding sales tax?

To solve this problem, let's use the 4-step problem solving model.

Mrs. Nguyen wants to apply fertilizer to her front lawn. A bag of fertilizer that covers 1,000 square feet costs $18. How many bags of fertilizer will Mrs. Nguyen need to purchase?

Surface Area Problem

After a storm, the Serafina family needs to have their roof replaced. Their house is in the shape of a pentagonal prism with the dimensions shown in the diagram.

To match their new roof, Mrs. Serafina decided to have both pentagonal sides of their house covered in aluminum siding. Their house is in the shape of a pentagonal prism with the dimensions shown in the diagram.

A contractor gave Mrs. Serafina an estimate based on a cost of $3.10 per square foot to complete the aluminum siding. How much will it cost the Serafina family to have the aluminum siding installed?

Solving Problems Using Proportionality

Proportional relationships are another important part of geometric problem solving.

A woodblock painting has dimensions of 60 centimeters by 79.5 centimeters. In order to fit on a flyer advertising the opening of a new art show, the image must be reduced by a scale factor of  1/25.

W hat will be the final dimensions of the image on the flyer?

Measuring Problem

For summer vacation, Jennifer and her family drove from their home in Inlandton to Beachville. Their car can drive 20 miles on one gallon of gasoline. Use the ruler to measure the distance that they drove to the nearest  1/4  inch, and then calculate the number of gallons of gasoline their car will use at this rate to drive from Inlandton to Beachville.

Practice #1

A blueprint for a rectangular tool shed has dimensions shown in the diagram below.

Todd is using this blueprint to build a tool shed, and he wants to surround the base of the tool shed with landscaping timbers as a border. How many feet of landscaping timbers will Todd need?

Practice #2

A scale model of a locomotive is shown. Use the ruler to measure the dimensions of the model to the nearest 1/4  inch, and then calculate the actual dimensions of the locomotive.

Scale : 1 inch = 5 feet

Solving geometric problems, such as those found in art and architecture, is an important skill. As with any mathematical problem, you can use the 4-step problem solving model to help you think through the important parts of the problem and be sure that you don't miss key information.

There are a lot of different applications of geometry to real-world problem solving. Some of the more common applications include the following:

What is the perimeter of the base of the cup, if the cup is in the shape of an octagonal prism?

The JP Morgan Chase Bank Tower in downtown Houston, Texas, is one of the tallest buildings west of the Mississippi River. It is in the shape of a pentagonal prism. If 40% of each face is covered with glass windows, what is the amount of surface area covered with glass?

The dimensions of Vincent van Gogh's Starry Night are 29 inches by 36 1 4 36\frac{1}{4} inches. If a print reduces these dimensions by a scale factor of 30%, what will be the dimensions of the print?

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Chapter 4: Inequalities

4.5 Geometric Word Problems

It is common to run into geometry-based word problems that look at either the interior angles, perimeter, or area of shapes. When looking at interior angles, the sum of the angles of any polygon can be found by taking the number of sides, subtracting 2, and then multiplying the result by 180°. In other words:

[latex]\text{sum of interior angles} = 180^{\circ} \times (\text{number of sides} - 2)[/latex]

This means the interior angles of a triangle add up to 180° × (3 − 2), or 180°. Any four-sided polygon will have interior angles adding to 180° × (4 − 2), or 360°. A chart can be made of these:

[latex]\begin{array}{rrrrrr} \text{3 sides:}&180^{\circ}&\times&(3-2)&=&180^{\circ} \\ \text{4 sides:}&180^{\circ}&\times&(4-2)&=&360^{\circ} \\ \text{5 sides:}&180^{\circ}&\times&(5-2)&=&540^{\circ} \\ \text{6 sides:}&180^{\circ}&\times&(6-2)&=&720^{\circ} \\ \text{7 sides:}&180^{\circ}&\times&(7-2)&=&900^{\circ} \\ \text{8 sides:}&180^{\circ}&\times&(8-2)&=&1080^{\circ} \\ \end{array}[/latex]

Example 4.5.1

The second angle [latex](A_2)[/latex] of a triangle is double the first [latex](A_1).[/latex] The third angle [latex](A_3)[/latex] is 40° less than the first [latex](A_1).[/latex] Find the three angles.

The relationships described in equation form are as follows:

[latex]A_2 = 2A_1 \text{ and } A_3 = A_1 - 40^{\circ}[/latex]

Because the shape in question is a triangle, the interior angles add up to 180°. Therefore:

[latex]A_1 + A_2 + A_3 = 180^{\circ}[/latex]

Which can be simplified using substitutions:

[latex]A_1 + (2A_1) + (A_1 - 40^{\circ}) = 180^{\circ}[/latex]

Which leaves:

[latex]\begin{array}{rrrrrrrrrrr} 2A_1&+&A_1&+&A_1&-&40^{\circ}&=&180^{\circ}&&&\\ &&&&4A_1&-&40^{\circ}&=&180^{\circ}&&\\ \\ &&&&&&4A_1&=&180^{\circ}&+&40^{\circ}\\ \\ &&&&&&A_1&=&\dfrac{220^{\circ}}{4}&\text{or}&55^{\circ} \end{array}[/latex]

This means [latex]A_2 = 2 (55^{\circ})[/latex] or 110° and [latex]A_3 = 55^{\circ}-40^{\circ}[/latex] or 15°.

Another common geometry word problem involves perimeter, or the distance around an object. For example, consider a rectangle, for which [latex]\text{perimeter} = 2l + 2w.[/latex]

Example 4.5.2

If the length of a rectangle is 5 m less than twice the width, and the perimeter is 44 m long, find its length and width.

[latex]L = 2W - 5 \text{ and } P = 44[/latex]

For a rectangle, the perimeter is defined by:

[latex]P = 2 W + 2 L[/latex]

Substituting for [latex]L[/latex] and the value for the perimeter yields:

[latex]44 = 2W + 2 (2W - 5)[/latex]

Which simplifies to:

[latex]44 = 2W + 4W - 10[/latex]

Further simplify to find the length and width:

[latex]\begin{array}{rrrrlrrrr} 44&+&10&=&6W&&&& \\ \\ &&54&=&6W&&&& \\ \\ &&W&=&\dfrac{54}{6}&\text{or}&9&& \\ \\ &\text{So}&L&=&2(9)&-&5&\text{or}&13 \\ \end{array}[/latex]

The width is 9 m and the length is 13 m.

Other common geometric problems are:

Example 4.5.3

A 15 m cable is cut into two pieces such that the first piece is four times larger than the second. Find the length of each piece.

[latex]P_1 + P_2 = 15 \text{ and } P_1 = 4P_2[/latex]

Combining these yields:

[latex]\begin{array}{rrrrrrr} 4P_2&+&P_2&=&15&& \\ \\ &&5P_2&=&15&& \\ \\ &&P_2&=&\dfrac{15}{5}&\text{or}&3 \end{array}[/latex]

This means that [latex]P_2 =[/latex] 3 m and [latex]P_1 = 4 (3),[/latex] or 12 m.

For questions 1 to 8, write the formula defining each relation. Do not solve.

• The length of a rectangle is 3 cm less than double the width, and the perimeter is 54 cm.
• The length of a rectangle is 8 cm less than double its width, and the perimeter is 64 cm.
• The length of a rectangle is 4 cm more than double its width, and the perimeter is 32 cm.
• The first angle of a triangle is twice as large as the second and 10° larger than the third.
• The first angle of a triangle is half as large as the second and 20° larger than the third.
• The sum of the first and second angles of a triangle is half the amount of the third angle.
• A 140 cm cable is cut into two pieces. The first piece is five times as long as the second.
• A 48 m piece of hose is to be cut into two pieces such that the second piece is 5 m longer than the first.

For questions 9 to 18, write and solve the equation describing each relationship.

• The second angle of a triangle is the same size as the first angle. The third angle is 12° larger than the first angle. How large are the angles?
• Two angles of a triangle are the same size. The third angle is 12° smaller than the first angle. Find the measure of the angles.
• Two angles of a triangle are the same size. The third angle is three times as large as the first. How large are the angles?
• The second angle of a triangle is twice as large as the first. The measure of the third angle is 20° greater than the first. How large are the angles?
• Find the dimensions of a rectangle if the perimeter is 150 cm and the length is 15 cm greater than the width.
• If the perimeter of a rectangle is 304 cm and the length is 40 cm longer than the width, find the length and width.
• Find the length and width of a rectangular garden if the perimeter is 152 m and the width is 22 m less than the length.
• If the perimeter of a rectangle is 280 m and the width is 26 m less than the length, find its length and width.
• A lab technician cuts a 12 cm piece of tubing into two pieces such that one piece is two times longer than the other. How long are the pieces?
• An electrician cuts a 30 m piece of cable into two pieces. One piece is 2 m longer than the other. How long are the pieces?

Answer Key 4.5

Intermediate Algebra Copyright © 2020 by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Scale Drawings - Grade 7

Suggested learning targets.

• I can solve problems involving scale drawings, when given scale.
• I can compute lengths and area from a scale drawing.
• I can reproduce the drawing of a different scale using a given scale drawing.

The following diagram shows an example of scale drawing. Scroll down the page for more examples and solutions on how to use scale drawings.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

o          Circle

o          Radius

o          Diameter

o          Arc

o          Subtend

o          Sector

o          Chord

o          Secant

o          Tangent

o          Circumference

o          Know how to fundamentally define a circle

o          Identify various parts of circles

o          Calculate the area and circumference of a circle or portion thereof

Introduction to Circles

A circle is simply a figure defined by all the points that are equidistant from a given center point. Thus, we can define a circle (without a definite location) by specifying the distance from the center; alternatively, we can define a circle (with a definite location) by specifying the distance from and the location of the center. The distance from the center of the points that lie on the circle is called the radius (plural radii ) of the circle. A circle with a radius r and center point C is shown below.

Note that regardless of which point on the circle is chosen, it is always a distance r from the center point, C. Each of the line segments drawn from C to the circle is called a radius (in other words, no one segment is defined as the radius, since all such segments are equal in length). The circle itself does not show any angles or sides that we can use to determine how many degrees are in the figure (as we did with polygons), but we can see that any two radii do form an angle α , as shown below.

Using our measurement of degrees, this angle α can (unambiguously) be any value between 0° and 360°. We can also define angles with either positive or negative numbers, depending on the direction of measurement from a particular radius--a positive angle is traditionally measured in the counterclockwise direction, and a negative angle is traditionally measured in the clockwise direction, as shown below.

We can also identify other parts of a circle. Given any point P 1 on the circle, the most distant point P 2 that is also on the circle is on the exact opposite side, and these points are the length of two radii apart. The line segment that connects them passes through the center and is called the diameter of the circle.

Any two radii in the circle (which form an angle α ), such as those shown below, form an arc and a sector. The arc is the portion of the circle opposite to (or subtended by) the angle α and between the endpoints of the radii. The sector is the area enclosed by the arc and the radii. The arc A in the diagram below is shown as a bold line, and the sector S is shown as a shaded area.

Any line segment connecting two points on the circle is called a chord . Note that a diameter is a chord (the longest possible chord of a circle!). The chord D is shown below as a bold line.

Other special figures related to circles are secants and tangents. A secant is simply a line that intersects two points of the circle (a chord is a segment of a secant). A tangent is a line that intersects the circle at exactly one point. Secant E and tangent T are shown in the diagram below as bold lines.

Practice Problem : Identify each part (labeled A through E) of the circle below.

Solution : Each portion of the circle or other line can be identified by its relationship to the entire circle. Each of these figures is discussed and defined above. The shaded area A is a sector. The line (or line segment) B is a tangent (note that it intersects the circle at only one point). C is a diameter, D is a chord, and E is a radius.

Basic Characteristics of Circles

Now that we have identified some of the components of circles, we can now begin to derive some of their characteristics using the tools we have developed thus far. Some of the properties of circles require trigonometry to derive (such as the length of a chord subtended by an angle α ), but others we can derive or simply state basic formulas that we can use to solve problems. Let's start with the circumference and area of a circle. The circumference of a circle is simply the length of the boundary (that is, the perimeter) of the circle. We will simply state the formula for the circumference C of a circle in terms of the radius ( r ) or the diameter ( d ):

C = 2 πr = πd

Note that with the formula for the circumference, we introduce the number π . Because π is (most mathematicians think) an irrational number, we cannot write it out exactly in its decimal form, nor can we write it exactly as a fraction. We can, however, write a decimal estimation of π that is sufficient for our purposes.

π ≈ 3.1416

Many calculators have a built-in key for the number π (although the calculator still just uses an approximation of π ). For many calculations, an approximate π value of 3.14 yields sufficient accuracy. As it turns out, π also appears in the formula for the area of a circle. Let's write the area A in terms of the radius r .

A = πr 2

Note again that we have not really derived these formulas; we are simply stating them as fundamental facts upon which we will base the rest of our investigation of the characteristics of circles.

Practice Problem : Find the area and circumference of a circle with a diameter of 4 inches.

Solution : One of the first rules of solving these types of problems involving circles is to carefully note whether we are dealing with the radius or the diameter. In this problem, the circle is described using the diameter, which is 4 inches. The radius is thus 2 inches. Let's now calculate the area A and circumference C using the formulas given above.

A = πr 2 = π (2 in) 2 ≈ (3.14)(4 in 2 ) =12.56 in 2

C = πd = π (4 in) ≈ (3.14) (4 in) = 12.56 in

Coincidentally, the area and circumference have the same numerical values (but not the same units!). This is not generally the case, of course.

Let's take a look at the characteristics of other parts of a circle. For instance, now that we know how to calculate the circumference of a circle, we can also calculate the length of an arc (which is simply a portion of the circumference). An angle α defined by two radii subtends an arc. Let's take a look at several examples, from which we can identify a pattern. The arc K in each case is shown as a bold curve. The circumference of the circle is C .

The expressions under each example can be derived by inspection. We know that if the angle α is 90° (one-quarter of the full 360° of the circle), then the subtended arc is one-quarter of the circumference. The same reasoning applies to determining the arc length K for the other two cases as well. Thus, we can see that the arc length is related to the circumference as the angle α is related to 360°. But this is simply a ratio that we can write as follows.

Thus, the area of sector S is related to the area A according to the ratio of α to 360°. We can once again derive a formula.

Practice Problem : A central angle γ in a circle of radius 10 units forms a sector with an area of 2.62 square units. Find the measure of γ .

Solution : Let's start by drawing a diagram of the problem. This diagram need not be to scale-we can simply use it to more easily identify the parts of the circle discussed in the problem. We'll call the sector S , and we know S = 2.62 units 2 .

We can use the formula presented earlier to relate the area S of the sector to the area A of the circle.

But we also know how to calculate A using the given radius of 10 units.

A = π r 2 = π(10 units) 2 ≈ 3.14(100 units 2 ) ≈ 314 units 2

Now let's find γ .

Thus, the angle has a measure of 3°.

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Grade 10 Mathematics Module: Circles and Other Geometric Figures on the Coordinate Plane

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

The SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pretest is provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the posttest to self-check your learning. Answer Key is provided for each activity and test. We trust that you will be honest in using these.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

This module was designed and written with you in mind. It is here to help you graph and solve problems involving circles and other geometric figures on the coordinate plane. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course but the pacing in which you read and answer this module is dependent on your ability.

After going through this module, you are expected to:

1. graph a circle and other geometric figures on the coordinate plane; and

2. solve problems involving circles and other geometric figures.

Grade 10 Mathematics Quarter 2 Self-Learning Module: Circles and Other Geometric Figures on the Coordinate Plane

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Area of Composite Figures Word Problems

The area of composite shapes is the area that is covered by any composite shape. The composite shape is a shape in which few polygons are put together to form a required shape. These shapes or figures can be made up of a combination of triangles, squares, and quadrilaterals, etc. Divide a composite shape into basic shapes like square, triangle, rectangle, hexagon, etc. to determine the area of composite shapes.

1. Read the word problem and decompose the composite figure into simpler shapes of which you can find the area. 2. Write down the area formulas of the simpler shapes. 3. Substitute the given values into the formulas. 4. Add or subtract the areas of the simple shapes as needed and state the correct square units.

The area is composed of a square and two semicircles (one whole circle).

Area of the pool = Area of a circle + Area of a square

= πr 2 + s 2

= 3.14 (18.25 m) 2 + (36.5 m) 2

= 1045.81 m 2 + 1332.25 m 2

= 2378.1 m 2

Practice Area of Composite Figures Word Problems

Practice Problem 1

Practice Problem 2

Practice Problem 3

Composite figure – a figure created by adding two or more shapes.

Area – the space occupied by a flat shape or the surface of an object.

Pre-requisite Skills Measurement – Perimeter Measurement – Area Circumference and Area of Circles Circles: Diameter, Radius, Circumference Area of a parallelogram Area of a triangle

Related Skills Surface Area of Prisms Surface Area of Pyramids Surface Area of Cylinders Surface Area of Cones Surface Area of Rectangular Prisms Surface Area of Composite Figures

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7.1.3: Triangles

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Learning Objectives

• Identify equilateral, isosceles, scalene, acute, right, and obtuse triangles.
• Identify whether triangles are similar, congruent, or neither.
• Identify corresponding sides of congruent and similar triangles.
• Find the missing measurements in a pair of similar triangles.
• Solve application problems involving similar triangles.

Introduction

Geometric shapes, also called figures, are an important part of the study of geometry. The triangle is one of the basic shapes in geometry. It is the simplest shape within a classification of shapes called polygons . All triangles have three sides and three angles, but they come in many different shapes and sizes. Within the group of all triangles, the characteristics of a triangle’s sides and angles are used to classify it even further. Triangles have some important characteristics, and understanding these characteristics allows you to apply the ideas in real-world problems.

Classifying and Naming Triangles

A polygon is a closed plane figure with three or more straight sides. Polygons each have a special name based on the number of sides they have. For example, the polygon with three sides is called a triangle because “tri” is a prefix that means “three.” Its name also indicates that this polygon has three angles. The prefix “poly” means many.

The table below shows and describes three classifications of triangles. Notice how the types of angles in the triangle are used to classify the triangle.

The sum of the measures of the three interior angles of a triangle is always 180 o . This fact can be applied to find the measure of the third angle of a triangle, if you are given the other two. Consider the examples below.

A triangle has two angles that measure 35 o and 75 o . Find the measure of the third angle.

The third angle of the triangle measures 70 o .

One of the angles in a right triangle measures 57 o . Find the measurement of the third angle.

The third angle of the right triangle measures 33 o .

There is an established convention for naming triangles. The labels of the vertices of the triangle, which are generally capital letters, are used to name a triangle.

You can call this triangle \(\ A B C\) or \(\ \triangle A B C\) since \(\ A\), \(\ B\), and \(\ C\) are vertices of the triangle. When naming the triangle, you can begin with any vertex. Then keep the letters in order as you go around the polygon. The triangle above could be named in a variety of ways: \(\ \triangle A B C\), or \(\ \triangle C B A\). The sides of the triangle are line segments \(\ A B\), \(\ AC\), and \(\ CB\).

Just as triangles can be classified as acute, obtuse, or right based on their angles, they can also be classified by the length of their sides. Sides of equal length are called congruent sides. While we designate a segment joining points \(\ A\) and \(\ B\) by the notation \(\ \overline{A B}\), we designate the length of a segment joining points \(\ A\) and \(\ B\) by the notation \(\ AB\) without a segment bar over it. The length \(\ AB\) is a number, and the segment \(\ \overline{A B}\) is the collection of points that make up the segment.

Mathematicians show congruency by putting a hash mark symbol through the middle of sides of equal length. If the hash mark is the same on one or more sides, then those sides are congruent. If the sides have different hash marks, they are not congruent. The table below shows the classification of triangles by their side lengths.

To describe a triangle even more specifically, you can use information about both its sides and its angles. Consider this example.

Classify the triangle below.

This is an isosceles right triangle.

Classify the triangle shown below.

• acute scalene
• right isosceles
• obtuse scalene
• obtuse isosceles
• Incorrect. This triangle has one angle (angle \(\ Q\)) that is between 90 o and 180 o , so it is an obtuse triangle. It is also scalene because all the sides have different lengths. The correct answer is obtuse scalene.
• Incorrect. This triangle does not contain a right angle. It has one angle (angle \(\ Q\)) that is somewhere between 90 o and 180 o , so it is an obtuse triangle. It is also scalene because all the sides have different lengths. The correct answer is obtuse scalene.
• Correct. This triangle has vertices \(\ P\), \(\ Q\), and \(\ R\), one angle (angle \(\ Q\)) that is between 90 o and 180 o , and sides of three different lengths.
• Incorrect. Although this triangle is obtuse, it does not have two sides of equal length. Its three sides are all different lengths, so it is scalene. The correct answer is obtuse scalene.

Identifying Congruent and Similar Triangles

Two triangles are congruent if they are exactly the same size and shape. In congruent triangles, the measures of corresponding angles and the lengths of corresponding sides are equal. Consider the two triangles shown below:

Since both \(\ \angle B\) and \(\ \angle E\) are right angles, these triangles are right triangles. Let’s call these two triangles \(\ \triangle A B C\) and \(\ \triangle D E F\). These triangles are congruent if every pair of corresponding sides has equal lengths and every pair of corresponding angles has the same measure.

The corresponding sides are opposite the corresponding angles.

A double headed arrow means “corresponds to”

\(\ \begin{array}{l} \angle B \leftrightarrow \angle E \\ \angle A \leftrightarrow \angle D \\ \angle C \leftrightarrow \angle F \\ \overline{A B} \leftrightarrow \overline{D E} \\ \overline{A C} \leftrightarrow \overline{D F} \\ \overline{B C} \leftrightarrow \overline{E F} \end{array}\)

\(\ \triangle A B C\) and \(\ \triangle D E F\) are congruent triangles as the corresponding sides and corresponding angles are equal.

Let’s take a look at another pair of triangles. Below are the triangles \(\ \triangle A B C\) and \(\ \triangle R S T\).

These two triangles are surely not congruent because \(\ \triangle R S T\) is clearly smaller in size than \(\ \triangle A B C\). But, even though they are not the same size, they do resemble one another. They are the same shape. The corresponding angles of these triangles look like they might have the same exact measurement, and if they did they would be congruent angles and we would call the triangles similar triangles.

Congruent angles are marked with hash marks, just as congruent sides are.

We can also show congruent angles by using multiple bands within the angle, rather than multiple hash marks on one band. Below is an image using multiple bands within the angle.

If the corresponding angles of two triangles have the same measurements, they are called similar triangles. This name makes sense because they have the same shape, but not necessarily the same size. When a pair of triangles is similar, the corresponding sides are proportional to one another. That means that there is a consistent scale factor that can be used to compare the corresponding sides. In the previous example, the side lengths of the larger triangle are all 1.4 times the length of the smaller. So, similar triangles are proportional to one another.

Just because two triangles look similar does not mean they are similar triangles in the mathematical sense of the word. Checking that the corresponding angles have equal measure is one way of being sure the triangles are similar.

Corresponding Sides of Similar Triangles

There is another method for determining similarity of triangles that involves comparing the ratios of the lengths of the corresponding sides.

If the ratios of the pairs of corresponding sides are equal, the triangles are similar.

Consider the two triangles below.

\(\ \triangle A B C\) is not congruent to \(\ \triangle D E F\) because the side lengths of \(\ \triangle D E F\) are longer than those of \(\ \triangle A B C\). So, are these triangles similar? If they are, the corresponding sides should be proportional.

Since these triangles are oriented in the same way, you can pair the left, right, and bottom sides: \(\ \overline{A B}\) and \(\ \overline{D E}\), \(\ \overline{B C}\) and \(\ \overline{E F}\), \(\ \overline{A C}\) and \(\ \overline{D E}\). (You might have called these the two shortest sides, the two longest sides, and the two leftover sides and arrived at the same ratios). Now we will look at the ratios of their lengths.

\(\ \frac{A B}{D E}=\frac{B C}{E F}=\frac{A C}{D F}\)

Substituting the side length values into the proportion, you see that it is true:

\(\ \frac{3}{9}=\frac{4}{12}=\frac{6}{18}\)

If the corresponding sides are proportional, then the triangles are similar. Triangles \(\ A B C\) and \(\ D E F\) are similar, but not congruent.

Let’s use this idea of proportional corresponding sides to determine whether two more triangles are similar.

Determine if the triangles below are similar by seeing if their corresponding sides are proportional.

\(\ \triangle A B C\) and \(\ \triangle D E F\) are similar.

The mathematical symbol ~ means “is similar to.” So, you can write \(\ \triangle A B C\) is similar to \(\ \triangle D E F\) as \(\ \triangle A B C \sim \triangle D E F\).

Determine whether the two triangles are similar, congruent, or neither.

• \(\ \triangle A B C \text { and } \triangle D E F\) are congruent.
• \(\ \triangle A B C \text { and } \triangle D E F\) are similar.
• \(\ \triangle A B C \text { and } \triangle D E F\) are similar and congruent.
• \(\ \triangle A B C \text { and } \triangle D E F\) are neither similar nor congruent.
• Incorrect. Congruent triangles have corresponding sides of equal length and corresponding angles of equal measure. They are the same exact size and shape. \(\ \triangle A B C\) is equilateral and \(\ \triangle D E F\) is isosceles, so they are not the same exact shape. The correct answer is \(\ \triangle A B C\) and \(\ \triangle D E F\) are neither similar nor congruent.
• Incorrect. The ratios of the corresponding sides are not equal, so the triangles cannot be similar: \(\ \frac{6.5}{5}=\frac{6.5}{5} \neq \frac{5}{5}\). The correct answer is \(\ \triangle A B C\) and \(\ \triangle D E F\) are neither similar nor congruent.
• Incorrect. All congruent triangles are similar, but these triangles are not congruent. Congruent triangles have corresponding sides of equal length and corresponding angles of equal measure. \(\ \triangle A B C\) is equilateral and \(\ \triangle D E F\) is isosceles, so they are not the same exact shape. The correct answer is \(\ \triangle A B C\) and \(\ \triangle D E F\) are neither similar nor congruent.
• Correct. The corresponding angle measures are not known to be equal as shown by the absence of congruence marks on the angles. Also, the ratios of the corresponding sides are not equal: \(\ \frac{6.5}{5}=\frac{6.5}{5} \neq \frac{5}{5}\).

Finding Missing Measurements in Similar Triangles

You can find the missing measurements in a triangle if you know some measurements of a similar triangle. Let’s look at an example.

\(\ \triangle A B C\) and \(\ \triangle X Y Z\) are similar triangles. What is the length of side \(\ B C\)?

The missing length of side \(\ BC\) is 8 units.

This process is fairly straightforward, but be careful that your ratios represent corresponding sides, recalling that corresponding sides are opposite corresponding angles.

Solving Application Problems Involving Similar Triangles

Applying knowledge of triangles, similarity, and congruence can be very useful for solving problems in real life. Just as you can solve for missing lengths of a triangle drawn on a page, you can use triangles to find unknown distances between locations or objects.

Let’s consider the example of two trees and their shadows. Suppose the sun is shining down on two trees, one that is 6 feet tall and the other whose height is unknown. By measuring the length of each shadow on the ground, you can use triangle similarity to find the unknown height of the second tree.

First, let’s figure out where the triangles are in this situation! The trees themselves create one pair of corresponding sides. The shadows cast on the ground are another pair of corresponding sides. The third side of these imaginary similar triangles runs from the top of each tree to the tip of its shadow on the ground. This is the hypotenuse of the triangle.

If you know that the trees and their shadows form similar triangles, you can set up a proportion to find the height of the tree.

When the sun is at a certain angle in the sky, a 6--foot tree will cast a 4-foot shadow. How tall is a tree that casts an 8-foot shadow?

The tree is 12 feet tall.

Triangles are one of the basic shapes in the real world. Triangles can be classified by the characteristics of their angles and sides, and triangles can be compared based on these characteristics. The sum of the measures of the interior angles of any triangle is 180 o . Congruent triangles are triangles of the same size and shape. They have corresponding sides of equal length and corresponding angles of the same measurement. Similar triangles have the same shape, but not necessarily the same size. The lengths of their sides are proportional. Knowledge of triangles can be a helpful in solving real-world problems.

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Definition:

Scale drawings are used to easily illustrate and compute the actual sizes of objects that are inconvenient to draw.

Scale drawing is commonly used by architects and engineers for planning and building their projects. Scale drawing is all about ratio of the drawing size to the actual size of the object. Thus, Drawing size : Actual size.

Ratio  is a comparison of two values or quantities. It indicates the relationship of the two quantities to one another.

Proportion  is also a comparison of two values that indicates that the two values are equivalent.

Examples of ratio:  5:8, 9:2, ½ : 6, ¾, ¼ Examples of proportion:  ½ = 4/8, ⅔ = 6/9, ¾ = 9/12, ⅓ = 2/6, ⅘ = 8/10

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