Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Lesson 8.2 Answer Key Converse of the Pythagorean Theorem.

## Texas Go Math Grade 8 Lesson 8.2 Answer Key Converse of the Pythagorean Theorem

**Texas Go Math Grade 8 Lesson 8.2 Explore Activity Answer Key**

**Testing the Converse of the Pythagorean Theorem**

The Pythagorean Theorem states that if a triangle is a right triangle, then a^{2} + b^{2} = c^{2}.

The converse of the Pythagorean Theorem states that if a^{2} + b^{2} = c^{2}, then the triangle is a right triangle.

Decide whether the converse of the Pythagorean Theorem is true.

**A.** Verify that the following sets of lengths make the equation a^{2} + b^{2} = c^{2} true. Record your results in the table.

**B.** For each set of lengths in the table, cut strips of grid paper with a width of one square and lengths that correspond to the values of a, b, and c.

**C.** For each set of lengths, use the strips of grid paper to try to form a right triangle. An example using the first set of lengths is shown. Record your findings in the table.

**Reflect**

Question 1.

Draw Conclusions Based on your observations, explain whether you think the converse of the Pythagorean Theorem is true.

Answer:

Based on our observations, we saw that the sets of lengths that made the equation a^{2} + b^{2} = c^{2 }true, also formed a right triangle. Therefore, the converse of the Pythagorean Theorem is true.

**Example 1**

Tell whether each triangle with the given side lengths is a right triangle.

**A.** 9 inches, 40 inches, and 41 inches

Let a = 9, b = 40, and c = 41.

Since 9^{2} + 40^{2} = 41^{2}, the triangle is a right triangle by the converse of the Pythagorean Theorem.

**B.** 8 meters, 10 meters, and 12 meters

Let a = 8, b = 10, and c = 12.

Since 8^{2} + 10^{2} ≠ 12^{2}, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

**Your Turn**

**Tell whether each triangle with the given side lengths is a right triangle.**

Question 2.

14 cm, 23 cm, and 25 cm

Answer:

The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. The Longest side is 25cm, so thats the hypothenuse c. Let a = 14cm and b = 23cm.

a^{2} + b^{2} = c^{2} …………. (1)

14^{2} + 23^{2} = 25^{2} (Substitute into the formula) ………….. (2)

196 + 529 = 625 (Simplify) ………… (3)

725 ≠ 625 (Add) ……….. (4)

Since 14^{2} + 23^{2} ≠ 252, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

Question 3.

16 in., 30 in., and 34 in.

Answer:

The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. The longest side is 34 in, so thats the hypothenuse c = 34 in. Let a = 30 in and b = 16 in.

a^{2} + b^{2} = c^{2} …………. (1)

30^{2} + 16^{2} = 34^{2} (Substitute into the formula) ………….. (2)

900 + 256 = 1156 (Simplify) ………… (3)

1156 = 1156 (Add) ……….. (4)

Since 30^{2} + 16^{2} = 34^{2}, the triangle is a right triangle by the converse of the Pythagorean Theorem.

Question 4.

27 ft, 36 ft, 45 ft

Answer:

The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. The longest side is the hypothenuse c = 45 cm. Let a = 27 cm and b = 36 cm.

a^{2} + b^{2} = c^{2} …………. (1)

27^{2} + 36^{2} = 45^{2} (Substitute into the formula) ………….. (2)

729 + 1296 = 2025 (Simplify) ………… (3)

2025 = 2025 (Add) ……….. (4)

Since 27^{2} + 36^{2} = 45^{2}, the triangle is a right triangle by the converse of the Pythagorean Theorem.

Question 5.

11 mm, 18 mm, 21 mm

Answer:

The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. Let a = 11 mm, b = 18 m and C = 2 1 mm.

a^{2} + b^{2} = c^{2} …………. (1)

11^{2} + 18^{2} = 21^{2} (Substitute into the formula) …………. (2)

121 + 324 = 441 (Simplify) ………… (3)

445 ≠ 441 (Add) ………………… (4)

Since 11^{2} + 18^{2} ≠ 21^{2}, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

**Example 2**

Katya is buying edging for a triangular flower garden she plans to build in her backyard. If the lengths of the three pieces of edging that she purchases are 13 feet, 10 feet, and 7 feet, will the flower garden be in the shape of a right triangle?

Use the converse of the Pythagorean Theorem. Remember to use the longest length for c.

Let a = 7, b = 10,and c = 13.

a^{2} + b^{2} = c^{2}

Since 7^{2} + 10^{2} ≠ 13^{2}, the garden will not be in the shape of a right triangle.

**Your Turn**

Question 6.

A blueprint for a new triangular playground shows that the sides measure 480 ft, 140 ft, and 500 ft. Is the playground in the shape of a right triangle? Explain.

Answer:

Let a = 480 ft, b = 140 ft and c = 500 ft

a^{2} + b^{2} = c^{2} …………. (1)

4802 + 1402 = 5002 (Substitute into the formula) ………… (2)

230400 + 19600 = 250000 (Simplify) ………… (3)

250000 = 250000 (Add) ………….. (4)

Since 480^{2} + 140^{2} = 500^{2} the playground is in the shape of a right triangle.

Question 7.

A triangular piece of glass has sides that measure 18 in., 19 in., and 25 in. Is the piece of glass in the shape of a right triangle? Explain.

Answer:

Let a = 18 in, b = 19 in and c = 25 in

a^{2} + b^{2} = c^{2} …………. (1)

18^{2} + 19^{2} = 25^{2} (Substitute into the formula) ………… (2)

324 + 361 = 625 (Simplify) ……………. (3)

685 ≠ 625 (Add) …………… (4)

Since 18^{2} + 19^{2} ≠ 25^{2}, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

The piece of glass is not in the shape of a right triangle.

Question 8.

A corner of a fenced yard forms a right angle. Can you place a 12 foot long board across the corner to form a right triangle for which the leg lengths are whole numbers? Explain.

Answer:

No, we cannot find two whole numbers to serve as leg lengths, such that the sum of their squares to be 144.

**Texas Go Math Grade 8 Lesson 8.2 Guided Practice Answer Key**

Question 1.

Lashandra used grid paper to construct the triangle shown. (Explore Activity)

a. What are the lengths of the sides of Lashandra’s triangle?

__________ units, __________ units, __________ units

Answer:

Lashandra used grid paper to construct the triangle which lengths of the sides are number of units.

Let count units: a = 8 units, b = 6 units, c = 10 units.

b. Use the converse of the Pythagorean Theorem to determine whether the triangle is a right triangle.

The triangle that Lashandra constructed (is/is not) a right triangle.

Answer:

Using the converse of the Pythagorean Theorem we can determine whether the triangle is a right triangle.

a^{2} + b^{2} = c^{2} …………. (1)

8^{2} + 6^{2} = 10^{2} (Substitute) …………….. (2)

64 + 36 = 100 (Simplify) ………………. (3)

100 = 100 (Add) …………… (4)

As 8^{2} + 6^{2} = 10^{2} the triangle is the right triangle.

Question 2.

A triangle has side lengths 9 cm, 12 cm, and 16 cm. Tell whether the triangle is a right triangle. (Example 1)

Let a = __________, b = __________, and c = __________.

By the converse of the Pythagorean Theorem, the triangle (is/is not) a right triangle.

Answer:

Let a= 9, b = 12 and c = 16.

a^{2} + b^{2} = c^{2} …………. (1)

9^{2} + 12^{2} = 16^{2} (Substitute into formula) ………….. (2)

81 + 144 = 256 (Simplify) ………….. (3)

225 ≠ 256 (Add) …………… (4)

By the converse of the Pythagorean Theorem, the triangle IS NOT a right triangle.

Question 3.

The marketing team at a new electronics company is designing a logo that contains a circle and a triangle. On one design, the triangle’s side lengths are 2.5 in., 6 in., and 6.5 in. Is the triangle a right triangle? Explain. (Example 2)

Answer:

Let a = 2.5in, b = 6m and c = 6.5in.

a^{2} + b^{2} = c^{2} …………. (1)

(2.5)^{2} + 6^{2} = (6.5)^{2} (Substitute into formula) ………….. (2)

6.25 + 36 = 42.25 (Simplify) ………………. (3)

42.25 = 42.25 (Add) ………………. (4)

By the converse of the Pythagorean Theorem, the triangle ( is / is not) a right triangle.

The triangle is a right triangle.

**Essential Question Check-In**

Question 4.

How can you use the converse of the Pythagorean Theorem to tell if a triangle is a right triangle?

Answer:

Knowing the side lengths, we substitute them in the formula a^{2} + b^{2} = c^{2}, where c contains the biggest value. If the equation holds true, then the given triangle is a right triangle. Otherwise, it is not a right triangle.

**Texas Go Math Grade 8 Lesson 8.2 Independent Practice Answer Key**

**Tell whether each triangle with the given side lengths is a right triangle.**

Question 5.

11 cm, 60 cm, 61 cm

Answer:

Let a = 11, b = 60 and c = 61. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

11^{2} + 60^{2} = 61^{2}

121 + 3600 = 3721

3721 = 3721

True

Since 11^{2} + 60^{2} = 61^{2}, the triangle is a right triangle.

Question 6.

5 ft, 12 ft, 15 ft

Answer:

Let a = 5, b = 12 and c = 15 Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

5^{2} + 12^{2} = 15^{2}

25 + 144 = 225

169 ≠ 225

False

Since 5^{2} + 12^{2} ≠ 152, the triangle is not a right triangle.

Question 7.

9 in., 15 in., 17 in.

Answer:

Let a = 9 in, b = 15 in, and C = 17 in.

a^{2} + b^{2} = c^{2} ……………….. (1)

9^{2} + 15^{2} = 17^{2} (Substitute into the formula) ………….. (2)

81 + 225 = 289 (Simplify) …………… (3)

306 ≠ 289 (Add) …………… (4)

Since 9^{2} + 15^{2}≠ 17^{2}, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

Question 8.

15 m, 36 m, 39 m

Answer:

Let a = 15m, b = 36m and C = 39m.

a^{2} + b^{2} = c^{2} ……………….. (1)

15^{2} + 36^{2} = 39^{2} (Substitute into the formula) ………….. (2)

225 + 1296 = 1521 (Simplify) ………… (3)

1521 = 1521 (Add) ………….. (4)

Since 15^{2} + 36^{2} = 39^{2}, the triangle is a right triangle by the converse of the Pythagorean Theorem.

Question 9.

20 mm, 30 mm, 40 mm

Answer:

Let a = 20mm, b = 30mm and c = 40mm.

a^{2} + b^{2} = c^{2} ……………….. (1)

20^{2} + 30^{2} = 40^{2 }(Substitute into the formula) …………. (2)

400 + 900 = 1600 (SimpLify) …………. (3)

1300 ≠ 1600 (Add) …………. (4)

Since 20^{2} + 30^{2} ≠ 40^{2}, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

Question 10.

20 cm, 48 cm, 52 cm

Answer:

Let a = 20 cm, b = 48 cm and c = 52 cm.

a^{2} + b^{2} = c^{2} ……………….. (1)

20^{2} + 48^{2} = 52^{2 } (Substitute into the formula) …………. (2)

400 + 2304 = 2704 (Simplify) …………… (3)

2704 = 2704 (Add) ……………. (4)

Since 20^{2} + 48^{2} = 52^{2}, the triangle is a right triangle by the converse of the Pythagorean Theorem.

Question 11.

18.5 ft, 6 ft, 17.5 ft

Answer:

Let a = 6ft, b = 17.5ft and c = 18.5ft.

a^{2} + b^{2} = c^{2} ……………….. (1)

6^{2} + (17.5)^{2} = (18.5)^{2} (Substitute into the formula) ……………. (2)

36 + 306.25 = 342.25 (Simplify) ………………. (3)

342.25 = 342.25 (Add) ………….. (4)

Since 6^{2} + (17.5)^{2} = (18.5)^{2}, the triangle is right triangle by the converse of the Pythagorean Theorem.

Question 12.

2 mi, 1.5 mi, 2.5 mi

Answer:

Let a = 2, b = 1.5 and c = 2.5. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

2^{2} + 1.5^{2} = 2.5^{2}

4 + 2.25 = 6.25

6.25 = 6.25

True

Since 2^{2} + 1.5^{2} = 2.5^{2}, the triangle is a right triangle.

Question 13.

35 in., 45 in., 55 in.

Answer:

Let a = 35 in, b = 45 in and c = 55 in.

a^{2} + b^{2} = c^{2} ……………….. (1)

35^{2} + 45^{2} = 55^{2} (Substitute into the formula) …………… (2)

1225 + 2025 = 3025 (Simplify) …………… (3)

3250 ≠ 3025 (Add) ……………. (4)

Since 35^{2} + 45^{2} ≠ 55^{2}, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

Question 14.

25cm, 14cm, 23cm

Answer:

Let a = 14, b = 23 and c = 25. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

14^{2} + 23^{2} = 25^{2}

196 + 529 = 625

725 = 625

False

Since 14^{2} + 23^{2} ≠ 25^{2}, the triangle is not a right triangle.

Question 15.

The emblem on a college banner consists of the face of a tiger inside a triangle. The lengths of the sides of the triangle are 13 cm, 14 cm, and 15 cm. Is the triangle a right triangle? Explain.

Answer:

Let a = 13 cm, b = 14 cm and c = 15 cm.

a^{2} + b^{2} = c^{2} ……………….. (1)

13^{2} + 14^{2} = 15^{2} (Substitute into the formula) ………………. (2)

169 + 196 = 225 (Simplify) ……………. (3)

365 ≠ 225 (Add) …………… (4)

Since 13^{2} + 14^{2} ≠ 15^{2}, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

Question 16.

Kerry has a large triangular piece of fabric that she wants to attach to the ceiling in her bedroom. The sides of the piece of fabric measure 4.8 ft, 6.4 ft, and 8 ft. Is the fabric in the shape of a right triangle? Explain.

Answer:

Let a = 4.8 ft, b = 6.4 ft and c = 8 ft

a^{2} + b^{2} = c^{2} ……………….. (1)

(4.8)^{2} + (6.4)^{2} = 8^{2 } (Substitute into the formula) ……………… (2)

23.04 + 40.96 = 64 (Simplify) ……………. (3)

64 = 64 (Add) ……………… (4)

Since (4.8)^{2} + (6.4)^{2} = 8^{2}, the triangle is a right triangle by the converse of the Pythagorean Theorem.

The piece of fabric in the shape of a right triangle.

Question 17.

A mosaic consists of triangular tiles. The smallest tiles have side lengths 6 cm, 10 cm, and 12 cm. Are these tiles in the shape of right triangles? Explain.

Answer:

Let a = 6, b = 10 and c = 12. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

6^{2} + 10^{2} = 12^{2}

36 + 100 = 144

136 = 144

False

Since 6^{2} + 10^{2} ≠ 12^{2}, the tiles are not in the shape of right triangles.

Question 18.

History In ancient Egypt, surveyors made right angles by stretching a rope with evenly spaced knots as shown. Explain why the rope forms a right angle.

Answer:

Every knote is placed on the equal distance, so the unit of measure ¡s part of the rope between two knotes. Let a = 4 units, b = 3 units, and c = 5 units.

a^{2} + b^{2} = c^{2} ……………….. (1)

4^{2} + 3^{2} = 5^{2} (Substitute into the formula) ……………… (2)

16 + 9 = 25 (Simplify) …………….. (3)

25 = 25 (Add) ……………… (4)

Since 4^{2} + 3^{2} = 5^{2}, the triangle is a right triangle by the converse of the Pythagorean Theorem.

Question 19.

**Justify Reasoning** Yoshi has two identical triangular boards as shown. Can he use these two boards to form a rectangle? Explain.

Answer:

If the triangle boards are right triangles, then Yoshi can form a rectangle. We can use the converse of the Pythagorean Theorem. Let a = 1 m, b = 0.75m and c = 1.25m

a^{2} + b^{2} = c^{2} ……………….. (1)

1^{2} + (0.75)^{2} = (1.25)^{2} (Substitute into the formula) ……………….. (2)

1 + 0.5625 = 1.5625 (Simplify) ……………… (3)

1.5625 = 1.5625 (Add) …………….. (4)

Since 1^{2} + (0.75)^{2} = (1.25)^{2}, triangle boards are right angled by the converse of the Pythagorean Theorem.

Yes, Yoshi can form a rectangle from those triangle boards.

Question 20.

**Critique Reasoning** Shoshanna says that a triangle with side lengths 17 m, 8 m, and 15 m is not a right triangle because 17^{2} + 8^{2} = 353, 15^{2} = 225, and 353 ≠ 225. Is she correct? Explain.

Answer:

The converse of the Pythagorean Theorem states that if a^{2} + b^{2} = c^{2} (where c is the biggest side length), then the triangle is a right triangle. Using this theorem, we see a = 8, b = 15 and c = 17.

a^{2} + b^{2} = c^{2}

8^{2} + 15^{2} = 17^{2}

64 + 225 = 289

289 = 289

True

Since 8^{2} + 15^{2} = 17^{2}, the triangle is a right triangle. Shoshanna was incorrect because she used the formula c^{2} + a^{2} = b^{2}, which is not the correct formula of the Pythagorean Theorem.

**H.O.T. Focus on Higher Order Thinking**

Question 21.

**Make a Conjecture** Diondre says that he can take any right triangle and make a new right triangle just by doubling the side lengths. Is Diondre’s conjecture true? Test his conjecture using three different right triangles.

Answer:

Given a right triangle, the Pythagorean Theorem holds. Therefore,

a^{2} + b^{2} = c^{2}

If we double the side lengths of that triangle, we get:

(2a)^{2} + (2b)^{2} = (2c)^{2}

4a^{2} + 4b^{2} = 4c^{2}

4(a^{2} + b^{2}2) = 4c^{2}

a^{2} + b^{2} = c^{2}

As we can see, doubling the side lengths of a right triangle would create a new right triangle.

We can test that by using three different right triangles.

The triangle with sides a = 3, b = 4 and c = 5 is a right triangle. We double its sides and check if the new triangle is a right triangle. We get a = 6, b = 8, c = 10.

6^{2} + 8^{2} = 10^{2}

36 + 64 = 100

100 = 100

True

Since 6^{2} + 8^{2} = 10^{2}, the new triangle is a right triangle by the converse of the Pythagorean Theorem.

The triangle with sides a = 6, b = 8 and c = 10 is a right triangle. We double its sides and check if the new triangle is a right triangle. We get a = 12, b = 16, c = 20.

12^{2} + 16^{2} = 20^{2}

144 + 256 = 400

400 = 400

True

Since 12^{2} + 16^{2} = 20^{2}, the new triangle is a right triangle by the converse of the Pythagorean Theorem.

The triangle with sides a = 12, b = 16 and c = 20 is a right triangle. We double its sides and check if the new triangle is a right triangle. We get a = 24, b = 32, c = 40.

24^{2} + 32^{2} = 40^{2}

576 + 1024 = 1600

1600 = 1600

True

Since 24^{2} + 32^{2} = 40^{2}, the new triangle is a right triangle by the converse of the Pythagorean Theorem.

Question 22.

**Draw Conclusions** A diagonal of a parallelogram measures 37 inches. The sides measure 35 inches and 1 foot. Is the parallelogram a rectangle? Explain your reasoning.

Answer:

A rectangle is a parallelogram where the interior angles are right angles. To prove if the given parallelogram is a rectangle, we need to prove that the triangle formed by the diagonal of the parallelogram and two sides of it, is a right triangle. Converting all the values into inches, we have a = 12, b = 35 and c = 37 Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

12^{2} + 35^{2} = 37^{2}

144 + 1225 = 1369

1369 = 1369

True

Since 12^{2} + 35^{2} = 37^{2}, the triangle is a right triangle. Therefore, the given parallelogram is a rectangle.

Question 23.

**Represent Real-World Problems** A soccer coach is marking the lines for a soccer field on a large recreation field. The dimensions of the field are to be 90 yards by 48 yards. Describe a procedure she could use to confirm that the sides of the field meet at right angles.

Answer:

To confirm that the sides of the field meet at right angles, she could measure the diagonaL of the fieLd and use the converse of the Pythagorean Theorem. If a^{2} + b^{2} = c^{2} (where a = 90, b = 48 and c is the length of the diagonal), then the triangle is a right triangle. This method can be used for every corner to decide if they form right angles or not.