i can apply the pythagorean theorem to solve real - world problems.
9. kevin is creating a target for a game that will be played at the school’s carnival and needs to put red tape in the shape of an “x” as shown below. if the area of the square board is 25 square feet, how many feet of tape will he need?
(there is a diagram of a square with its diagonals drawn)
10. gigi’s family left their house and drove 14 miles south to a gas station and 48 miles then trip to the water park. how much shorter would their trip have been if they hadn’t stopped at the gas station and driven along the diagonal path home?
(there is a right - triangle diagram with house, gas, and water park as vertices)
i can explain and use the pythagorean theorem converse.
11. is the triangle a right triangle? explain.
(there is a triangle with sides 6, 8, 12 and a right - angle mark)
12. is the triangle a right triangle? explain.
(there is a triangle with sides 12, 22.5, 25.5)
13. could the side lengths below form a right triangle? explain.
61, 60, 11
14. could the side lengths below form a right triangle? explain.
5, 7.5, 2.5
15. the triangular sail on esther’s sailboat measures 10.5 feet, 14 feet and 17.5 feet on its three sides. is the sail a right triangle? explain.
©koneversing the middle llc, 2017

Question

Accepted Answer

Let's solve these problems one by one using the Pythagorean theorem and its converse. The Pythagorean theorem states that in a right triangle, $ a^2 + b^2 = c^2 $, where $ c $ is the hypotenuse (the longest side), and $ a $ and $ b $ are the legs. The converse states that if $ a^2 + b^2 = c^2 $ for the side lengths of a triangle, then it is a right triangle.

### Problem 11: Is the triangle a right triangle? Explain.
Sides: $ 6 $, $ 8 $, $ 12 $ (hypotenuse should be the longest, so $ 12 $)
Check $ 6^2 + 8^2 = 12^2 $
$ 36 + 64 = 144 $
$ 100 = 144 $? No.
But wait, the triangle has a right angle symbol on the $ 6 $ and $ 8 $ sides. Wait, maybe I misread. Wait, the triangle has sides $ 6 $, $ 8 $, $ 12 $, but with a right angle between $ 6 $ and $ 8 $. Wait, no—if it's a right triangle, the legs squared should equal hypotenuse squared. Wait, $ 6^2 + 8^2 = 36 + 64 = 100 $, and $ 12^2 = 144 $. But the right angle symbol is on the triangle, so maybe the diagram is correct? Wait, maybe the sides are $ 6 $, $ 8 $, and hypotenuse $ 10 $, but here it's $ 12 $. Wait, maybe the problem has a typo, but according to the numbers:
$ 6^2 + 8^2 = 100 $, $ 12^2 = 144 $. Not equal. But the diagram shows a right angle. Wait, maybe the sides are $ 6 $, $ 8 $, $ 10 $, but labeled as $ 12 $? Wait, no, the given sides are $ 6 $, $ 8 $, $ 12 $. Wait, maybe the right angle is between $ 6 $ and $ 8 $, but then hypotenuse should be $ 10 $, not $ 12 $. So according to the numbers, $ 6^2 + 8^2
eq 12^2 $, so it's not a right triangle. But the diagram has a right angle—maybe the side lengths are mislabeled? Wait, maybe I made a mistake. Wait, $ 6^2 + 8^2 = 100 $, $ 12^2 = 144 $. So $ 100
eq 144 $, so it's not a right triangle. But the right angle symbol is there—maybe the problem is designed to check. Wait, maybe the sides are $ 6 $, $ 8 $, $ 10 $, but written as $ 12 $? No, the problem says $ 12 $. So:
$ 6^2 + 8^2 = 36 + 64 = 100 $
$ 12^2 = 144 $
$ 100
eq 144 $, so it is **not** a right triangle. But wait, the diagram has a right angle—maybe the side lengths are $ 6 $, $ 8 $, $ 10 $, but labeled as $ 12 $ by mistake? Alternatively, maybe I misread. Wait, the problem says "Is the triangle a right triangle? Explain." Given sides $ 6 $, $ 8 $, $ 12 $:
$ 6^2 + 8^2 = 100 $, $ 12^2 = 144 $. Not equal, so no.

### Problem 12: Is the triangle a right triangle? Explain.
Sides: $ 12 $, $ 22.5 $, $ 25.5 $ (hypotenuse is $ 25.5 $, longest)
Check $ 12^2 + 22.5^2 = 25.5^2 $
$ 144 + 506.25 = 650.25 $
$ 650.25 = 650.25 $? Yes!
$ 12^2 = 144 $, $ 22.5^2 = 506.25 $, sum is $ 650.25 $. $ 25.5^2 = (25 + 0.5)^2 = 25^2 + 2*25*0.5 + 0.5^2 = 625 + 25 + 0.25 = 650.25 $. So yes, it is a right triangle.

### Problem 13: Could the side lengths $ 61 $, $ 60 $, $ 11 $ form a right triangle? Explain.
Longest side: $ 61 $ (hypotenuse)
Check $ 11^2 + 60^2 = 61^2 $
$ 121 + 3600 = 3721 $
$ 3721 = 3721 $? Yes!
$ 11^2 = 121 $, $ 60^2 = 3600 $, sum is $ 3721 $. $ 61^2 = 3721 $. So yes, it is a right triangle.

### Problem 14: Could the side lengths $ 5 $, $ 7.5 $, $ 2.5 $ form a right triangle? Explain.
Longest side: $ 7.5 $ (hypotenuse)
Check $ 2.5^2 + 5^2 = 7.5^2 $
$ 6.25 + 25 = 56.25 $
$ 31.25 = 56.25 $? No.
$ 2.5^2 = 6.25 $, $ 5^2 = 25 $, sum is $ 31.25 $. $ 7.5^2 = 56.25 $. Not equal, so no.

### Problem 15: Is the sail a right triangle? Sides: $ 10.5 $, $ 14 $, $ 17.5 $
Longest side: $ 17.5 $ (hypotenuse)
Check $ 10.5^2 + 14^2 = 17.5^2 $
$ 110.25 + 196 = 306.25 $
$ 306.25 = 306.25 $? Yes!
$ 10.5^2 = 110.25 $, $ 14^2 = 196 $, sum is $ 306.25 $. $ 17.5^2 = (17 + 0.5)^2 = 17^2 + 2*17*0.5 + 0.5^2 = 289 + 17 + 0.25 = 306.25 $. So yes, it is a right triangle.

### Summary of Answers:
11. No, because $ 6^2 + 8^2
eq 12^2 $ (or if the right angle is mislabeled, but mathematically, the sides don’t satisfy the Pythagorean theorem).
12. Yes, because $ 12^2 + 22.5^2 = 25.5^2 $.
13. Yes, because $ 11^2 + 60^2 = 61^2 $.
14. No, because $ 2.5^2 + 5^2
eq 7.5^2 $.
15. Yes, because $ 10.5^2 + 14^2 = 17.5^2 $.

Let’s formalize each problem with step-by-step:

### 11. Step-by-Step:
## Step 1: Identify sides (legs and hypotenuse)
Legs: $ 6 $, $ 8 $; Hypotenuse: $ 12 $ (longest side).

## Step 2: Apply Pythagorean theorem
Check $ 6^2 + 8^2 = 12^2 $
$ 36 + 64 = 144 $
$ 100 = 144 $? No.

# Answer: No, because $ 6^2 + 8^2
eq 12^2 $.

### 12. Step-by-Step:
## Step 1: Identify sides
Legs: $ 12 $, $ 22.5 $; Hypotenuse: $ 25.5 $ (longest).

## Step 2: Check $ a^2 + b^2 = c^2 $
$ 12^2 + 22.5^2 = 144 + 506.25 = 650.25 $
$ 25.5^2 = 650.25 $

# Answer: Yes, because $ 12^2 + 22.5^2 = 25.5^2 $.

### 13. Step-by-Step:
## Step 1: Identify sides
Legs: $ 11 $, $ 60 $; Hypotenuse: $ 61 $ (longest).

## Step 2: Check $ a^2 + b^2 = c^2 $
$ 11^2 + 60^2 = 121 + 3600 = 3721 $
$ 61^2 = 3721 $

# Answer: Yes, because $ 11^2 + 60^2 = 61^2 $.

### 14. Step-by-Step:
## Step 1: Identify sides
Legs: $ 2.5 $, $ 5 $; Hypotenuse: $ 7.5 $ (longest).

## Step 2: Check $ a^2 + b^2 = c^2 $
$ 2.5^2 + 5^2 = 6.25 + 25 = 31.25 $
$ 7.5^2 = 56.25 $
$ 31.25 = 56.25 $? No.

# Answer: No, because $ 2.5^2 + 5^2
eq 7.5^2 $.

### 15. Step-by-Step:
## Step 1: Identify sides
Legs: $ 10.5 $, $ 14 $; Hypotenuse: $ 17.5 $ (longest).

## Step 2: Check $ a^2 + b^2 = c^2 $
$ 10.5^2 + 14^2 = 110.25 + 196 = 306.25 $
$ 17.5^2 = 306.25 $

# Answer: Yes, because $ 10.5^2 + 14^2 = 17.5^2 $.

### Final Answers:
11. No
12. Yes
13. Yes
14. No
15. Yes

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 11: Is the triangle a right triangle? Explain.

Problem 12: Is the triangle a right triangle? Explain.

Problem 13: Could the side lengths \( 61 \), \( 60 \), \( 11 \) form a right triangle? Explain.

Problem 14: Could the side lengths \( 5 \), \( 7.5 \), \( 2.5 \) form a right triangle? Explain.

Problem 15: Is the sail a right triangle? Sides: \( 10.5 \), \( 14 \), \( 17.5 \)

Answer:

12. Step-by-Step:

Step 1: Identify sides

Step 2: Check \( a^2 + b^2 = c^2 \)