8 2 assignment the pythagorean theorem and its converse

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• Pythagorean Theorem and its Converse

Key Questions

When carpenters want to construct a guaranteed right angle, they can make a triangle with sides 3, 4, and 5 (units). By the Pythagorean Theorem, a triangle made with these side lengths is always a right triangle, because #3^2 + 4^2 = 5^2.#

If you want to find out the distance between two places, but you only have their coordinates (or how many blocks apart they are), the Pythagorean Theorem says the square of this distance is equal to the sum of the squared horizontal and vertical distances. #d^2 = (x_1 - x_2)^2 + (y_1 - y_2)^2#

Say one place is at #(2,4)# and the other is at #(3, 1)# . (These could also be latitude and longitudes, but you get the idea.) Then we square the horizontal distance: #(2 - 3)^2 = 1#

and the vertical distance:

#(4 - 1)^2 = 9#

add these squares,

#1 + 9 = 10#

and then take the square root.

#d = sqrt10#

• TV sizes are measured on the diagonal; it gives the longest screen measurement. You can figure out what size TV can fit in a space by using the Pythagorean Theorem:

#("TV size")^2 = ("space width")^2 + ("space height")^2# Note: you should also remember that TVs are usually #16 xx 9,# so you'd likely want to measure just the width of the space, then use #"width "xx9/16# as the height of the space.

Pythagorean theorem: #a^2+b^2=c^2#

This only applies to right angle triangles (one of the angles is #90^@# s)

If you want to find the length of a triangle where it's A=3 B=4 C=?

you do #3^2+4^2=?#

you can also use it to find A or B by doing #C^2-A^2=B^2# or #C^2-B^2=A^2#

Proof of Converse of Pythagoras Theorem

Statement: In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Proof: Here, we are given a triangle ABC in which AC 2 = AB 2 + BC 2 . We need to prove that ∠B = 90°.

To start with, we construct a ΔPQR right-angled at Q such that PQ = AB and QR = BC.

Now, from Δ PQR, we have:

PR 2 = PQ 2 + QR 2 (Pythagoras Theorem, as ∠Q=90°)

or PR 2 = AB 2 + BC 2 (By construction)........ (1)

But AC 2 = AB 2 + BC 2 (Given).......... (2)

So, AC = PR (From (1) and (2)).............. (3)

Now, in ΔABC and ΔPQR,

AB = PQ (By construction)

BC = QR (By construction)

AC = PR (Proved in (3))

So, ΔABC ≃ ΔPQR (According to the SSS congruence)

∠B = ∠Q (Corresponding angles of congruent triangles)

∠Q = 90° (By construction)

So ∠B = 90°.

Hence, the converse of the Pythagoras theorem is proved.

Converse of Pythagoras Theorem Formula

The converse of Pythagoras theorem formula is c 2 = a 2 + b 2 , where a, b, and c are the sides of the triangle.

Related Topics

Listed below are a few topics related to the converse of the Pythagoras theorem, take a look.

• Right Triangle Formulas
• Hypotenuse Leg Theorem
• What is Similarity
• Similarity in Triangles

Examples on Converse of Pythagoras Theorem

Example 1: The side of the triangle is of length 8 units, 10 units, and 6 units. Is this triangle a right triangle? If so, which side is the hypotenuse?

We know that the hypotenuse is the longest side in a triangle. The side or lengths is given as 8 units, 10 units, and 6 units. Therefore, 10 units is the hypotenuse.

Using the converse of Pythagoras theorem, we get,

(10) 2 = (8) 2 + (6) 2

100 = 64 + 36

Since both sides are equal, the triangle is a right triangle.

Example 2: Check if the triangle is acute, right, or an obtuse triangle with side lengths as 6, 8, and 11 units.

Solution: According to the length, we know that 11 units are the longest side.

Compare the square lengths of both the sides in the equation c 2 = a 2 + b 2 .

(11) 2 = (6) 2 + (8) 2

121 = 36 + 64

Hence, (11) 2 > (6) 2 + (8) 2

Therefore, according to the application of converse of Pythagoras theorem (If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle), the triangle is an obtuse triangle.

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Practice Questions on Converse of Pythagoras Theorem

Faqs on converse of pythagoras theorem.

The coverse of the Pythagoras theore m states that, in a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

What is the Formula for Converse of Pythagoras Theorem?

The converse of Pythagoras theorem formula is c 2 = a 2 + b 2 , where a, b, and c are the sides of the triangle .

How Do You Prove the Converse of Pythagoras Theorem?

The converse of the Pythagoras theorem states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If we consider two triangles ΔABC and ΔPQR where c 2 = a 2 + b 2 then we can say ∠C is a right triangle.

What is the Converse of the Pythagoras Theorem Useful For?

The converse of the Pythagoras theorem is useful in determining if a triangle is a right triangle or not. Whereas, a Pythagorean theorem helps in determining the length of the missing side of a right triangle.

What is the Application of the Converse of Pythagoras Theorem?

The application of the converse of the Pythagoras theorem is that the measurements help in determining what type of a triangle it is. There are three scenarios that we can determine, they are:

• If the sum of the squares of two sides of a triangle is considered equivalent to the square of the hypotenuse, the triangle is a right triangle.
• If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle.
• If the sum of the squares of two sides of a triangle is greater than the square of the hypotenuse, the triangle is an acute triangle.

What is the Pythagorean Theorem?

The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. We can say, (Opposite) 2 + (Adjacent) 2 = (Hypotenuse) 2 ​​​​​​.

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8.2: The Pythagorean Theorem

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• Converse Of Pythagoras Theorem

Converse of Pythagoras Theorem

The converse of Pythagoras theorem  states that “If the square of a side is equal to the sum of the square of the other two sides, then triangle must be right angle triangle”. Whereas Pythagorean theorem states that the sum of the square of two sides (legs) is equal to the square of the hypotenuse of a right-angle triangle. But, in the reverse of the Pythagorean theorem, it is said that if this relation satisfies, then triangle must be right angle triangle. So, if the sides of a triangle have length, a, b and c and satisfy given condition a 2 + b 2 = c 2 , then the triangle is a right-angle triangle.

Let us see the proof of this theorem along with examples.

Converse of Pythagoras Theorem Proof

Statement: If the length of a triangle is a, b and c and c 2 = a 2 + b 2 , then the triangle is a right-angle triangle.

Proof: Construct another triangle, △EGF, such as AC = EG = b and BC = FG = a.

In △EGF, by Pythagoras Theorem:

EF 2 = EG 2 + FG 2 = b 2 + a 2 …………(1)

In △ABC, by Pythagoras Theorem:

AB 2 = AC 2 + BC 2 = b 2 + a 2 …………(2)

From equation (1) and (2), we have;

EF 2 = AB 2

⇒ △ ACB ≅ △EGF (By SSS postulate)

⇒ ∠G is right angle

Thus, △EGF is a right triangle.

Hence, we can say that the converse of Pythagorean theorem also holds.

Hence Proved.

As per the converse of the Pythagorean theorem, the formula for a right-angled triangle is given by:

Where a, b and c are the sides of a triangle.

Applications

Basically, the converse of the Pythagoras theorem is used to find whether the measurements of a given triangle belong to the right triangle or not. If we come to know that the given sides belong to a right-angled triangle, it helps in the construction of such a triangle. Using the concept of the converse of Pythagoras theorem, one can determine if the given three sides form a Pythagorean triplet.

Converse of Pythagoras Theorem Examples

Question 1: The sides of a triangle are 5, 12 and 13. Check whether the given triangle is a right triangle or not?

Solution: Given,

By using the converse of Pythagorean Theorem,

a 2 +b 2 = c 2

c 2 = a 2 +b 2

Substitute the given values in the above equation,

13 2 = 5 2 +12 2

169 = 25 + 144

So, the given lengths are does not satisfy the above condition.

Therefore, the given triangle is a right triangle.

Question 2: The sides of a triangle are 7, 11 and 13. Check whether the given triangle is a right triangle or not?

Solution: Given;

Substitute the given values in the the above equation,

13 2 = 7 2 + 11 2

169 = 49 + 121

So, it is not satisfied with the above condition.

Therefore, the given triangle is not a right triangle.

Question 3: The sides of a triangle are 4,6 and 8. Say whether the given triangle is a right triangle or not.

Solution: Given: a = 4, b = 6, c = 8

By the converse of Pythagoras theorem

8 2 = 4 2 + 6 2

64 = 16 + 36

The sides of the given triangle do not satisfy the condition a 2 +b 2 = c 2 .

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Geometry 8-1 Complete Lesson: The Pythagorean Theorem and Its Converse

• The hypotenuse is the longest side in a right triangle.
• The hypotenuse of a right triangle can be any one of the three sides.
• In a triangle with side lengths 3, 4, and 5, one of the legs measures 4.

• One leg of the triangle has length 9 cm.
• The hypotenuse of the triangle has length 12 cm.
• The side lengths of the triangle form a Pythagorean triple.

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