7. a, 65 cm; c, 95 cm 8. a, 16 yd; b, 22 yd determine whether each triangle with sides of given lengths is a right triangle. justify your answer. 9. 18 ft, 23 ft, 29 ft 10. 7 yd, 24 yd, 25 yd 11. the hypotenuse of a right triangle is 15 inches, and one of its legs is 11 inches. find the length of the other leg. 12. a leg of a right triangle is 30 meters long, and the hypotenuse is 35 meters long. what is the length of the other leg? 13. televisions the diagonal of a television measures 27 inches. if the width of a 27 - inch is 22 inches, calculate its height to the nearest inch.

Question

Accepted Answer

Let's solve these problems one by one using the Pythagorean theorem ($a^2 + b^2 = c^2$, where $c$ is the hypotenuse, and $a, b$ are the legs).

### Problem 7: $ a = 65 \, \text{cm}, c = 95 \, \text{cm} $ (Find $ b $)
# Explanation:
## Step 1: Recall the Pythagorean theorem
The Pythagorean theorem states $ a^2 + b^2 = c^2 $. We need to solve for $ b $, so rearrange the formula: $ b^2 = c^2 - a^2 $.

## Step 2: Substitute the given values
Substitute $ a = 65 $ and $ c = 95 $ into the formula:
$ b^2 = 95^2 - 65^2 $
Calculate $ 95^2 = 9025 $ and $ 65^2 = 4225 $.

## Step 3: Simplify the equation
$ b^2 = 9025 - 4225 = 4800 $

## Step 4: Take the square root of $ b^2 $
$ b = \sqrt{4800} \approx 69.28 \, \text{cm} $ (rounded to two decimal places)

# Answer:
$ b \approx 69.28 \, \text{cm} $

### Problem 8: $ a = 16 \, \text{yd}, b = 22 \, \text{yd} $ (Find $ c $)
# Explanation:
## Step 1: Recall the Pythagorean theorem
$ a^2 + b^2 = c^2 $

## Step 2: Substitute the given values
Substitute $ a = 16 $ and $ b = 22 $:
$ c^2 = 16^2 + 22^2 $
Calculate $ 16^2 = 256 $ and $ 22^2 = 484 $.

## Step 3: Simplify the equation
$ c^2 = 256 + 484 = 740 $

## Step 4: Take the square root of $ c^2 $
$ c = \sqrt{740} \approx 27.20 \, \text{yd} $ (rounded to two decimal places)

# Answer:
$ c \approx 27.20 \, \text{yd} $

### Problem 9: Determine if $ 18 \, \text{ft}, 23 \, \text{ft}, 29 \, \text{ft} $ is a right triangle
# Explanation:
## Step 1: Identify the hypotenuse (longest side)
The longest side is $ 29 \, \text{ft} $, so assume $ c = 29 $, $ a = 18 $, $ b = 23 $.

## Step 2: Apply the Pythagorean theorem
Check if $ a^2 + b^2 = c^2 $.
Calculate $ 18^2 = 324 $, $ 23^2 = 529 $, $ 29^2 = 841 $.

## Step 3: Sum the squares of the legs
$ 324 + 529 = 853 $

## Step 4: Compare to the square of the hypotenuse
$ 853
eq 841 $, so $ a^2 + b^2
eq c^2 $.

# Answer:
No, the triangle with sides $ 18 \, \text{ft}, 23 \, \text{ft}, 29 \, \text{ft} $ is not a right triangle.

### Problem 10: Determine if $ 7 \, \text{yd}, 24 \, \text{yd}, 25 \, \text{yd} $ is a right triangle
# Explanation:
## Step 1: Identify the hypotenuse (longest side)
The longest side is $ 25 \, \text{yd} $, so assume $ c = 25 $, $ a = 7 $, $ b = 24 $.

## Step 2: Apply the Pythagorean theorem
Check if $ a^2 + b^2 = c^2 $.
Calculate $ 7^2 = 49 $, $ 24^2 = 576 $, $ 25^2 = 625 $.

## Step 3: Sum the squares of the legs
$ 49 + 576 = 625 $

## Step 4: Compare to the square of the hypotenuse
$ 625 = 625 $, so $ a^2 + b^2 = c^2 $.

# Answer:
Yes, the triangle with sides $ 7 \, \text{yd}, 24 \, \text{yd}, 25 \, \text{yd} $ is a right triangle.

### Problem 11: Hypotenuse $ c = 15 \, \text{in} $, leg $ a = 11 \, \text{in} $ (Find leg $ b $)
# Explanation:
## Step 1: Rearrange the Pythagorean theorem
$ b^2 = c^2 - a^2 $

## Step 2: Substitute the values
$ b^2 = 15^2 - 11^2 $
Calculate $ 15^2 = 225 $, $ 11^2 = 121 $.

## Step 3: Simplify
$ b^2 = 225 - 121 = 104 $

## Step 4: Take the square root
$ b = \sqrt{104} \approx 10.20 \, \text{in} $ (rounded to two decimal places)

# Answer:
$ b \approx 10.20 \, \text{in} $

### Problem 12: Leg $ a = 30 \, \text{m} $, hypotenuse $ c = 35 \, \text{m} $ (Find leg $ b $)
# Explanation:
## Step 1: Rearrange the Pythagorean theorem
$ b^2 = c^2 - a^2 $

## Step 2: Substitute the values
$ b^2 = 35^2 - 30^2 $
Calculate $ 35^2 = 1225 $, $ 30^2 = 900 $.

## Step 3: Simplify
$ b^2 = 1225 - 900 = 325 $

## Step 4: Take the square root
$ b = \sqrt{325} \approx 18.03 \, \text{m} $ (rounded to two decimal places)

# Answer:
$ b \approx 18.03 \, \text{m} $

### Problem 13: Television diagonal $ c = 27 \, \text{in} $, width $ a = 22 \, \text{in} $ (Find height $ b $)
# Explanation:
## Step 1: Rearrange the Pythagorean theorem
$ b^2 = c^2 - a^2 $

## Step 2: Substitute the values
$ b^2 = 27^2 - 22^2 $
Calculate $ 27^2 = 729 $, $ 22^2 = 484 $.

## Step 3: Simplify
$ b^2 = 729 - 484 = 245 $

## Step 4: Take the square root and round to the nearest inch
$ b = \sqrt{245} \approx 15.65 \approx 16 \, \text{in} $ (rounded to the nearest inch)

# Answer:
The height is approximately $ 16 \, \text{in} $.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 7: \( a = 65 \, \text{cm}, c = 95 \, \text{cm} \) (Find \( b \))

Step 1: Recall the Pythagorean theorem

Step 2: Substitute the given values

Step 3: Simplify the equation

Step 4: Take the square root of \( b^2 \)

Step 1: Recall the Pythagorean theorem

Step 2: Substitute the given values

Step 3: Simplify the equation

Step 4: Take the square root of \( c^2 \)

Step 1: Identify the hypotenuse (longest side)

Step 2: Apply the Pythagorean theorem

Step 3: Sum the squares of the legs

Step 4: Compare to the square of the hypotenuse

Answer:

Problem 8: \( a = 16 \, \text{yd}, b = 22 \, \text{yd} \) (Find \( c \))