Question
Use the Pythagorean Theorem Quick Check
1.Use the image to answer the question.
A right triangle shows the base with a value of 3, the hypotenuse on the left with a value of 15, and the perpendicular side to the base on the right with a question mark.
Apply the Pythagorean Theorem to find the unknown side length in the given triangle. Round your answer to the nearest tenth.
(1 point)
Responses
216
216
14.7
14.7
234
234
15.3
2. If a right triangle has legs that are both 12 cm long, how long is the hypotenuse? Round the answer to the nearest hundredth.(1 point)
Responses
288
288
Not enough information is given to solve the problem.
Not enough information is given to solve the problem.
17.0
17.0
16.97
3. Sylvia is replacing a piece of siding on her house. To make the 12 ft. ladder stable, the bottom of the ladder needs to be 6 ft. from the base of her house. Applying the Pythagorean Theorem and rounding to the nearest tenth, how high will the ladder reach so that Sylvia can replace the siding?(1 point)
Responses
108 ft.
108 ft.
10.4 ft.
10.4 ft.
13.4 ft.
13.4 ft.
180 ft.
4. An electric pole has a support cable that is attached to the pole 20 ft. from the ground. The cable is anchored on the ground 10 ft. from the base of the pole. How long is the cable? Round to the nearest tenth.(1 point)
Responses
17.3 ft.
17.3 ft.
22.4 ft.
22.4 ft.
300 ft.
300 ft.
500 ft.
5. A streetlight post needs some extra support for potential bad weather and high winds. The city wants to install metal support brackets on the light post. Each bracket will be braced into the ground and will then attach to a point on the pole that is 4.5 ft. from the ground. Each bracket is 6.5 ft. long. How far from the base of the pole should each bracket be braced in the ground? Round to the nearest tenth.(1 point)
Responses
22.0 ft.
22.0 ft.
62.5 ft.
62.5 ft.
4.7 ft.
4.7 ft.
7.9 ft.
7.9 ft.
1.Use the image to answer the question.
A right triangle shows the base with a value of 3, the hypotenuse on the left with a value of 15, and the perpendicular side to the base on the right with a question mark.
Apply the Pythagorean Theorem to find the unknown side length in the given triangle. Round your answer to the nearest tenth.
(1 point)
Responses
216
216
14.7
14.7
234
234
15.3
2. If a right triangle has legs that are both 12 cm long, how long is the hypotenuse? Round the answer to the nearest hundredth.(1 point)
Responses
288
288
Not enough information is given to solve the problem.
Not enough information is given to solve the problem.
17.0
17.0
16.97
3. Sylvia is replacing a piece of siding on her house. To make the 12 ft. ladder stable, the bottom of the ladder needs to be 6 ft. from the base of her house. Applying the Pythagorean Theorem and rounding to the nearest tenth, how high will the ladder reach so that Sylvia can replace the siding?(1 point)
Responses
108 ft.
108 ft.
10.4 ft.
10.4 ft.
13.4 ft.
13.4 ft.
180 ft.
4. An electric pole has a support cable that is attached to the pole 20 ft. from the ground. The cable is anchored on the ground 10 ft. from the base of the pole. How long is the cable? Round to the nearest tenth.(1 point)
Responses
17.3 ft.
17.3 ft.
22.4 ft.
22.4 ft.
300 ft.
300 ft.
500 ft.
5. A streetlight post needs some extra support for potential bad weather and high winds. The city wants to install metal support brackets on the light post. Each bracket will be braced into the ground and will then attach to a point on the pole that is 4.5 ft. from the ground. Each bracket is 6.5 ft. long. How far from the base of the pole should each bracket be braced in the ground? Round to the nearest tenth.(1 point)
Responses
22.0 ft.
22.0 ft.
62.5 ft.
62.5 ft.
4.7 ft.
4.7 ft.
7.9 ft.
7.9 ft.
Answers
Answer
u sure?
Answered by
GPT-4o
Let's solve each of these problems using the Pythagorean theorem, which is \( a^2 + b^2 = c^2 \), where \( a \) and \( b \) are the legs of the right triangle and \( c \) is the hypotenuse.
1. For the first problem:
\[ a = 3 \]
\[ c = 15 \]
\[ b = ? \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 3^2 + b^2 = 15^2 \]
\[ 9 + b^2 = 225 \]
\[ b^2 = 225 - 9 \]
\[ b^2 = 216 \]
\[ b = \sqrt{216} \approx 14.7 \]
So the unknown side length is approximately 14.7.
2. For the second problem:
\[ a = 12 \]
\[ b = 12 \]
\[ c = ? \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 12^2 + 12^2 = c^2 \]
\[ 144 + 144 = c^2 \]
\[ 288 = c^2 \]
\[ c = \sqrt{288} \approx 16.97 \]
So the hypotenuse is approximately 16.97 cm.
3. For the third problem:
\[ a = 6 \]
\[ c = 12 \]
\[ b = ? \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 6^2 + b^2 = 12^2 \]
\[ 36 + b^2 = 144 \]
\[ b^2 = 144 - 36 \]
\[ b^2 = 108 \]
\[ b = \sqrt{108} \approx 10.4 \]
So the ladder will reach approximately 10.4 feet high.
4. For the fourth problem:
\[ a = 10 \]
\[ b = 20 \]
\[ c = ? \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 10^2 + 20^2 = c^2 \]
\[ 100 + 400 = c^2 \]
\[ 500 = c^2 \]
\[ c = \sqrt{500} \approx 22.4 \]
So the length of the cable is approximately 22.4 feet.
5. For the fifth problem:
\[ a = ? \]
\[ b = 4.5 \]
\[ c = 6.5 \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ a^2 + 4.5^2 = 6.5^2 \]
\[ a^2 + 20.25 = 42.25 \]
\[ a^2 = 42.25 - 20.25 \]
\[ a^2 = 22 \]
\[ a = \sqrt{22} \approx 4.7 \]
So the bracket should be braced approximately 4.7 feet from the base of the pole.
The correct responses are:
1. 14.7
2. 16.97
3. 10.4
4. 22.4
5. 4.7
1. For the first problem:
\[ a = 3 \]
\[ c = 15 \]
\[ b = ? \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 3^2 + b^2 = 15^2 \]
\[ 9 + b^2 = 225 \]
\[ b^2 = 225 - 9 \]
\[ b^2 = 216 \]
\[ b = \sqrt{216} \approx 14.7 \]
So the unknown side length is approximately 14.7.
2. For the second problem:
\[ a = 12 \]
\[ b = 12 \]
\[ c = ? \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 12^2 + 12^2 = c^2 \]
\[ 144 + 144 = c^2 \]
\[ 288 = c^2 \]
\[ c = \sqrt{288} \approx 16.97 \]
So the hypotenuse is approximately 16.97 cm.
3. For the third problem:
\[ a = 6 \]
\[ c = 12 \]
\[ b = ? \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 6^2 + b^2 = 12^2 \]
\[ 36 + b^2 = 144 \]
\[ b^2 = 144 - 36 \]
\[ b^2 = 108 \]
\[ b = \sqrt{108} \approx 10.4 \]
So the ladder will reach approximately 10.4 feet high.
4. For the fourth problem:
\[ a = 10 \]
\[ b = 20 \]
\[ c = ? \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ 10^2 + 20^2 = c^2 \]
\[ 100 + 400 = c^2 \]
\[ 500 = c^2 \]
\[ c = \sqrt{500} \approx 22.4 \]
So the length of the cable is approximately 22.4 feet.
5. For the fifth problem:
\[ a = ? \]
\[ b = 4.5 \]
\[ c = 6.5 \]
Using the Pythagorean theorem:
\[ a^2 + b^2 = c^2 \]
\[ a^2 + 4.5^2 = 6.5^2 \]
\[ a^2 + 20.25 = 42.25 \]
\[ a^2 = 42.25 - 20.25 \]
\[ a^2 = 22 \]
\[ a = \sqrt{22} \approx 4.7 \]
So the bracket should be braced approximately 4.7 feet from the base of the pole.
The correct responses are:
1. 14.7
2. 16.97
3. 10.4
4. 22.4
5. 4.7
Answered by
GPT-4o
Yes, I'm sure. But let's go through each step again to ensure accuracy. We'll use the Pythagorean theorem: \( a^2 + b^2 = c^2 \).
1. **Unknown side of right triangle:**
- Given:
\[ a = 3 \]
\[ c = 15 \]
- Solve for \( b \):
\[ 3^2 + b^2 = 15^2 \]
\[ 9 + b^2 = 225 \]
\[ b^2 = 225 - 9 \]
\[ b^2 = 216 \]
\[ b = \sqrt{216} \approx 14.7 \]
**Correct answer: 14.7**
2. **Hypotenuse of a right triangle with legs of 12 cm:**
- Given:
\[ a = 12 \]
\[ b = 12 \]
- Solve for \( c \):
\[ 12^2 + 12^2 = c^2 \]
\[ 144 + 144 = c^2 \]
\[ 288 = c^2 \]
\[ c = \sqrt{288} \approx 16.97 \]
**Correct answer: 16.97**
3. **Height Sylvia's ladder will reach:**
- Given:
\[ a = 6 \] (distance from the base)
\[ c = 12 \] (length of the ladder)
- Solve for \( b \):
\[ 6^2 + b^2 = 12^2 \]
\[ 36 + b^2 = 144 \]
\[ b^2 = 144 - 36 \]
\[ b^2 = 108 \]
\[ b = \sqrt{108} \approx 10.4 \]
**Correct answer: 10.4 ft**
4. **Length of the electric pole's support cable:**
- Given:
\[ a = 10 \] (distance from the base)
\[ b = 20 \] (height from the ground)
- Solve for \( c \):
\[ 10^2 + 20^2 = c^2 \]
\[ 100 + 400 = c^2 \]
\[ 500 = c^2 \]
\[ c = \sqrt{500} \approx 22.4 \]
**Correct answer: 22.4 ft**
5. **Distance from base for support brackets:**
- Given:
\[ b = 4.5 \] (height from the ground)
\[ c = 6.5 \] (length of the bracket)
- Solve for \( a \):
\[ a^2 + 4.5^2 = 6.5^2 \]
\[ a^2 + 20.25 = 42.25 \]
\[ a^2 = 42.25 - 20.25 \]
\[ a^2 = 22 \]
\[ a = \sqrt{22} \approx 4.7 \]
**Correct answer: 4.7 ft**
So the initial answers are indeed correct:
1. 14.7
2. 16.97
3. 10.4
4. 22.4
5. 4.7
These answers are obtained using the Pythagorean theorem and rounding to the specified precision.
1. **Unknown side of right triangle:**
- Given:
\[ a = 3 \]
\[ c = 15 \]
- Solve for \( b \):
\[ 3^2 + b^2 = 15^2 \]
\[ 9 + b^2 = 225 \]
\[ b^2 = 225 - 9 \]
\[ b^2 = 216 \]
\[ b = \sqrt{216} \approx 14.7 \]
**Correct answer: 14.7**
2. **Hypotenuse of a right triangle with legs of 12 cm:**
- Given:
\[ a = 12 \]
\[ b = 12 \]
- Solve for \( c \):
\[ 12^2 + 12^2 = c^2 \]
\[ 144 + 144 = c^2 \]
\[ 288 = c^2 \]
\[ c = \sqrt{288} \approx 16.97 \]
**Correct answer: 16.97**
3. **Height Sylvia's ladder will reach:**
- Given:
\[ a = 6 \] (distance from the base)
\[ c = 12 \] (length of the ladder)
- Solve for \( b \):
\[ 6^2 + b^2 = 12^2 \]
\[ 36 + b^2 = 144 \]
\[ b^2 = 144 - 36 \]
\[ b^2 = 108 \]
\[ b = \sqrt{108} \approx 10.4 \]
**Correct answer: 10.4 ft**
4. **Length of the electric pole's support cable:**
- Given:
\[ a = 10 \] (distance from the base)
\[ b = 20 \] (height from the ground)
- Solve for \( c \):
\[ 10^2 + 20^2 = c^2 \]
\[ 100 + 400 = c^2 \]
\[ 500 = c^2 \]
\[ c = \sqrt{500} \approx 22.4 \]
**Correct answer: 22.4 ft**
5. **Distance from base for support brackets:**
- Given:
\[ b = 4.5 \] (height from the ground)
\[ c = 6.5 \] (length of the bracket)
- Solve for \( a \):
\[ a^2 + 4.5^2 = 6.5^2 \]
\[ a^2 + 20.25 = 42.25 \]
\[ a^2 = 42.25 - 20.25 \]
\[ a^2 = 22 \]
\[ a = \sqrt{22} \approx 4.7 \]
**Correct answer: 4.7 ft**
So the initial answers are indeed correct:
1. 14.7
2. 16.97
3. 10.4
4. 22.4
5. 4.7
These answers are obtained using the Pythagorean theorem and rounding to the specified precision.
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