Question

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

14.7
14.7
15.3
15.3
216
216
234

Answer 1

[AI-generated answer — Bot GPT 3.5]

To apply the Pythagorean Theorem, we can use the formula:

a^2 + b^2 = c^2

where a and b are the lengths of the two legs of the triangle, and c is the length of the hypotenuse.

In this case, the base (a) has a value of 3, the hypotenuse (c) has a value of 15, and we are trying to find the length of the perpendicular side (b).

We can set up the equation as follows:

3^2 + b^2 = 15^2

9 + b^2 = 225

b^2 = 216

Taking the square root of both sides to find b:

b ≈ 14.7

Therefore, the length of the unknown side is approximately 14.7.

Answer 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

17.0
17.0
Not enough information is given to solve the problem.
Not enough information is given to solve the problem.
16.97
16.97
288

Answer 3

[AI-generated answer — Bot GPT 3.5]

To find the length of the hypotenuse in a right triangle with legs of equal length, we can use the Pythagorean Theorem. The formula is:

c^2 = a^2 + b^2

where c is the length of the hypotenuse, and a and b are the lengths of the legs.

In this case, both legs have a length of 12 cm. Plugging the values into the formula:

c^2 = 12^2 + 12^2
c^2 = 144 + 144
c^2 = 288

Taking the square root of both sides to find c:

c ≈ √288
c ≈ 16.97

Therefore, the length of the hypotenuse is approximately 16.97 cm.

Answer 4

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

180 ft.
180 ft.
108 ft.
108 ft.
10.4 ft.
10.4 ft.
13.4 ft.
13.4 ft.

Answer 5

[AI-generated answer — Bot GPT 3.5]

To find the height that the ladder will reach, we can use the Pythagorean Theorem. The formula is:

a^2 + b^2 = c^2

where a is the distance from the base of the ladder to the house, b is the height that the ladder will reach, and c is the length of the ladder.

In this case, the distance from the base of the ladder to the house (a) is 6 ft, and the length of the ladder (c) is 12 ft.

Plugging the values into the formula:

6^2 + b^2 = 12^2
36 + b^2 = 144
b^2 = 144 - 36
b^2 = 108

Taking the square root of both sides to find b:

b ≈ √108
b ≈ 10.4

Therefore, the ladder will reach a height of approximately 10.4 ft.

Answer 6

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

4.7 ft.
4.7 ft.
7.9 ft.
7.9 ft.
22.0 ft.
22.0 ft.
62.5 ft.

Answer 7

[AI-generated answer — Bot GPT 3.5]

To find how far from the base of the pole each bracket should be braced in the ground, we can use the Pythagorean Theorem. The formula is:

a^2 + b^2 = c^2

where a is the height at which the brackets are attached to the pole (4.5 ft), b is the distance from the base of the pole to where each bracket is braced in the ground (unknown), and c is the length of each bracket (6.5 ft).

Plugging the values into the formula:

4.5^2 + b^2 = 6.5^2
20.25 + b^2 = 42.25
b^2 = 42.25 - 20.25
b^2 = 22

Taking the square root of both sides to find b:

b ≈ √22
b ≈ 4.7

Therefore, each bracket should be braced in the ground approximately 4.7 ft from the base of the pole.

Answer 8

Use the Converse of the Pythagorean Theorem to determine whether a right triangle can be formed given sides a, b, and c, where a=6, b=10, and c=12.(1 point)

Responses

No, a right triangle cannot be formed because 62+102≠122.
No, a right triangle cannot be formed because 6 squared plus 10 squared is not equal to 12 squared .
No, a right triangle cannot be formed because 62+102=122.
No, a right triangle cannot be formed because 6 squared plus 10 squared equals 12 squared .
Yes, a right triangle can be formed because 62+102=122.
Yes, a right triangle can be formed because 6 squared plus 10 squared equals 12 squared .
Yes, a right triangle can be formed because the Pythagorean Theorem produced a false statement.

Answer 9

[AI-generated answer — Bot GPT 3.5]

To determine if a right triangle can be formed with sides of length a=6, b=10, and c=12, we can use the converse of the Pythagorean Theorem. The converse states that if the sum of the squares of the two shorter sides is equal to the square of the longest side, then a right triangle can be formed.

In this case, we have a=6, b=10, and c=12.

Plugging the values into the converse of the Pythagorean Theorem:

6^2 + 10^2 = 12^2
36 + 100 = 144

Since 36 + 100 does not equal 144, we can conclude that a right triangle cannot be formed with sides of length a=6, b=10, and c=12.

Therefore, the correct response is: No, a right triangle cannot be formed because 6 squared plus 10 squared is not equal to 12 squared.