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8.2: Converse of the Pythagorean Theorem

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

• Understand the converse of the Pythagorean Theorem.
• Determine if a triangle is acute or obtuse from side measures.

Review Queue

1. Determine if the following sets of numbers are Pythagorean triples.
1. 14, 48, 50
2. 9, 40, 41
3. 12, 43, 44
4. 12, 35, 37
1. (512)2\begin{align*}\left( 5 \sqrt{12} \right )^2\end{align*}
2. 142\begin{align*}\frac{14}{\sqrt 2}\end{align*}
3. 183\begin{align*}\frac{18}{\sqrt 3}\end{align*}

Know What? A friend of yours is designing a building and wants it to be rectangular. One wall 65 ft. long and the other is 72 ft. long. How can he ensure the walls are going to be perpendicular?

Converse of the Pythagorean Theorem

Pythagorean Theorem Converse: If the square 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 a2+b2=c2\begin{align*}a^2 + b^2 = c^2\end{align*}, then ABC\begin{align*}\triangle ABC\end{align*} is a right triangle.

With this converse, you can use the Pythagorean Theorem to prove that a triangle is a right triangle, even if you do not know any angle measures.

Example 1: Determine if the triangles below are right triangles.

a)

b)

Solution: Check to see if the three lengths satisfy the Pythagorean Theorem. Let the longest side represent c\begin{align*}c\end{align*}.

a) a2+b2=c282+162=?(85)264+256=?645320=320Yes\begin{align*}a^2 + b^2 = c^2\!\\ 8^2 + 16^2 \overset{?}= \left( 8 \sqrt{5} \right )^2\!\\ 64 + 256 \overset{?}= 64 \cdot 5\!\\ 320 = 320 \qquad \text{Yes}\end{align*}

b) a2+b2=c2222+242=?262484+576=?6761060676No\begin{align*}a^2+b^2 = c^2\!\\ 22^2 + 24^2 \overset{?}= 26^2\!\\ 484 + 576 \overset{?}= 676\!\\ 1060 \neq 676 \qquad \text{No}\end{align*}

Example 2: Do the following lengths make a right triangle?

a) 5,3,14\begin{align*}\sqrt{5}, 3, \sqrt{14}\end{align*}

b) 6,23,8\begin{align*}6, 2 \sqrt{3}, 8\end{align*}

c) 32,42,52\begin{align*}3 \sqrt{2}, 4 \sqrt{2}, 5\sqrt{2}\end{align*}

Solution: Even though there is no picture, you can still use the Pythagorean Theorem. Again, the longest length will be c\begin{align*}c\end{align*}.

a) (5)2+32=1425+9=14Yes\begin{align*}\left( \sqrt{5} \right )^2 + 3^2 = \sqrt{14}^2\!\\ 5+9=14\!\\ \text{Yes}\end{align*}

b) 62+(23)2=8236+(43)=6436+1264\begin{align*}6^2 + \left( 2 \sqrt{3} \right )^2 = 8^2\!\\ 36+(4 \cdot 3) = 64\!\\ 36+12 \neq 64\end{align*}

c) This is a multiple of 2\begin{align*}\sqrt{2}\end{align*} of a 3, 4, 5 right triangle. Yes, this is a right triangle.

Identifying Acute and Obtuse Triangles

We can extend the converse of the Pythagorean Theorem to determine if a triangle is an obtuse or acute triangle.

Theorem 8-3: If the sum of the squares of the two shorter sides in a right triangle is greater than the square of the longest side, then the triangle is acute.

b<c\begin{align*}b < c\end{align*} and a<c\begin{align*}a < c\end{align*}

If a2+b2>c2\begin{align*}a^2 + b^2 > c^2\end{align*}, then the triangle is acute.

Theorem 8-4: If the sum of the squares of the two shorter sides in a right triangle is less than the square of the longest side, then the triangle is obtuse.

b<c\begin{align*}b < c\end{align*} and a<c\begin{align*}a < c\end{align*}

If a2+b2<c2\begin{align*}a^2+b^2, then the triangle is obtuse.

Example 3: Determine if the following triangles are acute, right or obtuse.

a)

b)

Solution: Set the longest side equal to c\begin{align*}c\end{align*}.

a) 62+(35)2 ? 8236+45 ? 6481>64\begin{align*}6^2 + \left( 3 \sqrt{5} \right)^2 \ ? \ 8^2\!\\ 36 + 45 \ ? \ 64\!\\ 81 > 64\end{align*}

The triangle is acute.

b) 152+142 ? 212225+196 ? 441421<441\begin{align*}15^2 + 14^2 \ ? \ 21^2\!\\ 225 + 196 \ ? \ 441\!\\ 421 < 441\end{align*}

The triangle is obtuse.

Example 4: Graph A(4,1),B(3,8)\begin{align*}A(-4, 1), B(3, 8)\end{align*}, and C(9,6)\begin{align*}C(9, 6)\end{align*}. Determine if ABC\begin{align*}\triangle ABC\end{align*} is acute, obtuse, or right.

Solution: Use the distance formula to find the length of each side.

\begin{align*}AB &= \sqrt{(-4-3)^2 + (1-8)^2} = \sqrt{49+49} = \sqrt{98} = 7 \sqrt{2}\\ BC &= \sqrt{(3-9)^2 + (8-6)^2} = \sqrt{36 + 4} = \sqrt{40} = 2 \sqrt{10}\\ AC &= \sqrt{(-4-9)^2 + (1-6)^2} = \sqrt{169 + 25} = \sqrt{194}\end{align*}

Plug these lengths into the Pythagorean Theorem.

\begin{align*}\left( \sqrt{98} \right )^2 + \left( \sqrt{40} \right)^2 & \ ? \ \left ( \sqrt{194} \right )^2\\ 98 + 40 & \ ? \ 194\\ 138 & < 194\end{align*}

\begin{align*}\triangle ABC\end{align*} is an obtuse triangle.

Know What? Revisited Find the length of the diagonal.

\begin{align*}65^2 + 72^2 &= c^2\\ 4225 + 5184 &= c^2\\ 9409 &= c^2\\ 97 &= c && \text{To make the building rectangular, both diagonals must be 97 feet.}\end{align*}

Review Questions

• Questions 1-6 are similar to Examples 1 and 2.
• Questions 7-15 are similar to Example 3.
• Questions 16-20 are similar to Example 4.
• Questions 21-24 use the Pythagorean Theorem.
• Question 25 uses the definition of similar triangles.

Determine if the following lengths make a right triangle.

1. 7, 24, 25
2. \begin{align*}\sqrt{5}, 2 \sqrt{10}, 3 \sqrt{5}\end{align*}
3. \begin{align*}2 \sqrt{3}, \sqrt{6}, 8\end{align*}
4. 15, 20, 25
5. 20, 25, 30
6. \begin{align*}8 \sqrt{3}, 6, 2 \sqrt{39}\end{align*}

Determine if the following triangles are acute, right or obtuse.

1. 7, 8, 9
2. 14, 48, 50
3. 5, 12, 15
4. 13, 84, 85
5. 20, 20, 24
6. 35, 40, 51
7. 39, 80, 89
8. 20, 21, 38
9. 48, 55, 76

Graph each set of points and determine if \begin{align*}\triangle ABC\end{align*} is acute, right, or obtuse, using the distance formula.

1. \begin{align*}A(3, -5), B(-5, -8), C(-2, 7)\end{align*}
2. \begin{align*}A(5, 3), B(2, -7), C(-1, 5)\end{align*}
3. \begin{align*}A(1, 6) , B(5, 2), C(-2, 3)\end{align*}
4. \begin{align*}A(-6, 1), B(-4, -5), C(5, -2)\end{align*}
5. Show that #18 is a right triangle by using the slopes of the sides of the triangle. The figure to the right is a rectangular prism. All sides (or faces) are either squares (the front and back) or rectangles (the four around the middle). All faces are perpendicular.
6. Find \begin{align*}c\end{align*}.
7. Find \begin{align*}d\end{align*}.

Now, the figure is a cube, where all the sides are squares. If all the sides have length 4, find:

1. Find \begin{align*}c\end{align*}.
2. Find \begin{align*}d\end{align*}.
3. Writing Explain why \begin{align*}m \angle A = 90^{\circ}\end{align*}.

1. Yes
2. Yes
3. No
4. Yes
1. \begin{align*}\left ( 5 \sqrt{12} \right )^2 = 5^2 \cdot 12 = 25 \cdot 12=300\end{align*}
2. \begin{align*}\frac{14}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{14 \sqrt{2}}{2} = 7 \sqrt{2}\end{align*}
3. \begin{align*}\frac{18}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}= \frac{18 \sqrt{3}}{3} = 6 \sqrt{3}\end{align*}

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