pythagorean theorem
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find the missing side of each triangle. leave your answers in simplest radical form.
1)
4 ft
x
15 ft
2)
5√2 mi
x
7 mi
3)
6 cm
x
11 cm
4)
16 cm
x
8 cm
5)
10 in
4 in
x
6)
2 in
√7 in
x
state if each triangle is a right triangle.
7)
√89 km
√170 km
9 km
8)
15 yd
2√26 yd
14 yd
9)
12 mi
8√2 mi
4 mi
10)
5 km
√11 km
6 km

Question

Accepted Answer

### Problem 1: Find the missing side (right triangle, legs 4 ft, $ x $, hypotenuse 15 ft)
# Explanation:
## Step1: Apply Pythagorean Theorem ($ a^2 + b^2 = c^2 $)
$ 4^2 + x^2 = 15^2 $
## Step2: Simplify and solve for $ x $
$ 16 + x^2 = 225 $
$ x^2 = 225 - 16 = 209 $
$ x = \sqrt{209} $ ft (since $ \sqrt{209} $ cannot be simplified further)
# Answer: $ \sqrt{209} $ ft

### Problem 2: Find the missing side (right triangle, legs $ 5\sqrt{2} $ mi, 7 mi, hypotenuse $ x $)
# Explanation:
## Step1: Apply Pythagorean Theorem
$ (5\sqrt{2})^2 + 7^2 = x^2 $
## Step2: Simplify and solve for $ x $
$ 25 \times 2 + 49 = x^2 $
$ 50 + 49 = x^2 $
$ x^2 = 99 $
$ x = \sqrt{99} = 3\sqrt{11} $ mi (simplify $ \sqrt{99} = \sqrt{9 \times 11} = 3\sqrt{11} $)
# Answer: $ 3\sqrt{11} $ mi

### Problem 3: Find the missing side (right triangle, legs 6 cm, 11 cm, hypotenuse $ x $)
# Explanation:
## Step1: Apply Pythagorean Theorem
$ 6^2 + 11^2 = x^2 $
## Step2: Simplify and solve for $ x $
$ 36 + 121 = x^2 $
$ x^2 = 157 $
$ x = \sqrt{157} $ cm (since $ \sqrt{157} $ is prime)
# Answer: $ \sqrt{157} $ cm

### Problem 4: Find the missing side (right triangle, hypotenuse 16 cm, leg 8 cm, leg $ x $)
# Explanation:
## Step1: Apply Pythagorean Theorem ($ a^2 + b^2 = c^2 $, solve for $ b $)
$ 8^2 + x^2 = 16^2 $
## Step2: Simplify and solve for $ x $
$ 64 + x^2 = 256 $
$ x^2 = 256 - 64 = 192 $
$ x = \sqrt{192} = 8\sqrt{3} $ cm (simplify $ \sqrt{192} = \sqrt{64 \times 3} = 8\sqrt{3} $)
# Answer: $ 8\sqrt{3} $ cm

### Problem 5: Find the missing side (right triangle, legs 10 in, 4 in, hypotenuse $ x $)
# Explanation:
## Step1: Apply Pythagorean Theorem
$ 10^2 + 4^2 = x^2 $
## Step2: Simplify and solve for $ x $
$ 100 + 16 = x^2 $
$ x^2 = 116 $
$ x = \sqrt{116} = 2\sqrt{29} $ in (simplify $ \sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29} $)
# Answer: $ 2\sqrt{29} $ in

### Problem 6: Find the missing side (right triangle, legs 2 in, $ \sqrt{7} $ in, hypotenuse $ x $)
# Explanation:
## Step1: Apply Pythagorean Theorem
$ 2^2 + (\sqrt{7})^2 = x^2 $
## Step2: Simplify and solve for $ x $
$ 4 + 7 = x^2 $
$ x^2 = 11 $
$ x = \sqrt{11} $ in
# Answer: $ \sqrt{11} $ in

### Problem 7: State if the triangle is right (sides $ \sqrt{89} $ km, 9 km, $ \sqrt{170} $ km)
# Explanation:
## Step1: Identify the longest side (hypotenuse candidate: $ \sqrt{170} \approx 13.04 $, $ \sqrt{89} \approx 9.43 $, 9)
## Step2: Check Pythagorean Theorem ($ a^2 + b^2 = c^2 $)
$ (\sqrt{89})^2 + 9^2 = 89 + 81 = 170 $
$ (\sqrt{170})^2 = 170 $
Since $ 89 + 81 = 170 $, the triangle satisfies the theorem.
# Answer: Yes, it is a right triangle.

### Problem 8: State if the triangle is right (sides 15 yd, $ 2\sqrt{26} $ yd, 14 yd)
# Explanation:
## Step1: Identify the longest side (hypotenuse candidate: 15 yd, $ 2\sqrt{26} \approx 10.19 $, 14)
## Step2: Check Pythagorean Theorem ($ a^2 + b^2 = c^2 $)
$ (2\sqrt{26})^2 + 14^2 = 4 \times 26 + 196 = 104 + 196 = 300 $
$ 15^2 = 225 $
Since $ 300
eq 225 $, the triangle does not satisfy the theorem.
# Answer: No, it is not a right triangle.

### Problem 9: State if the triangle is right (sides 12 mi, $ 8\sqrt{2} $ mi, 4 mi)
# Explanation:
## Step1: Identify the longest side (hypotenuse candidate: 12 mi, $ 8\sqrt{2} \approx 11.31 $, 4)
## Step2: Check Pythagorean Theorem ($ a^2 + b^2 = c^2 $)
$ (8\sqrt{2})^2 + 4^2 = 128 + 16 = 144 $
$ 12^2 = 144 $
Since $ 128 + 16 = 144 $, the triangle satisfies the theorem.
# Answer: Yes, it is a right triangle.

### Problem 10: State if the triangle is right (sides 5 km, $ \sqrt{11} $ km, 6 km)
# Explanation:
## Step1: Identify the longest side (hypotenuse candidate: 6 km, 5, $ \sqrt{11} \approx 3.31 $)
## Step2: Check Pythagorean Theorem ($ a^2 + b^2 = c^2 $)
$ 5^2 + (\sqrt{11})^2 = 25 + 11 = 36 $
$ 6^2 = 36 $
Since $ 25 + 11 = 36 $, the triangle satisfies the theorem.
# Answer: Yes, it is a right triangle.

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QUESTION IMAGE

Question

Explanation:

Problem 1: Find the missing side (right triangle, legs 4 ft, \( x \), hypotenuse 15 ft)

Step1: Apply Pythagorean Theorem (\( a^2 + b^2 = c^2 \))

Step2: Simplify and solve for \( x \)

Step1: Apply Pythagorean Theorem

Step2: Simplify and solve for \( x \)

Step1: Apply Pythagorean Theorem

Step2: Simplify and solve for \( x \)

Answer:

Problem 2: Find the missing side (right triangle, legs \( 5\sqrt{2} \) mi, 7 mi, hypotenuse \( x \))