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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
234
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9 answers

To find the unknown side length of the right triangle using the Pythagorean Theorem, we can apply the formula:

\[ a^2 + b^2 = c^2 \]

where:

  • \( a \) is one leg (base), which is 3,
  • \( b \) is the other leg (the unknown side),
  • \( c \) is the hypotenuse, which is 15.

Substituting the known values, the equation becomes:

\[ 3^2 + b^2 = 15^2 \]

Calculating the squares:

\[ 9 + b^2 = 225 \]

Now, to solve for \( b^2 \):

\[ b^2 = 225 - 9 \] \[ b^2 = 216 \]

Next, we take the square root of both sides to find \( b \):

\[ b = \sqrt{216} \] \[ b \approx 14.7 \]

So, rounding to the nearest tenth, the unknown side length is approximately 14.7.

The correct response is:

14.7

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

16.97
16.97

288
288

Not enough information is given to solve the problem.
Not enough information is given to solve the problem.

17.0

To find the length of the hypotenuse of a right triangle with legs that are both 12 cm long, we can use the Pythagorean Theorem:

\[ a^2 + b^2 = c^2 \]

where \( a \) and \( b \) are the legs of the triangle, and \( c \) is the hypotenuse.

Substituting the values we have:

\[ 12^2 + 12^2 = c^2 \]

Calculating the squares:

\[ 144 + 144 = c^2 \]

Adding them together:

\[ 288 = c^2 \]

Now, we take the square root of both sides to find \( c \):

\[ c = \sqrt{288} \]

Calculating the square root:

\[ c \approx 16.97 \]

Thus, rounding to the nearest hundredth, the length of the hypotenuse is approximately 16.97 cm.

The correct response is:

16.97

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

10.4 ft.
10.4 ft.

13.4 ft.
13.4 ft.

180 ft.
180 ft.

108 ft.

To find out how high the ladder will reach on the house, we can use the Pythagorean Theorem, which states:

\[ a^2 + b^2 = c^2 \]

where:

  • \( a \) is the height the ladder reaches (the side we want to find),
  • \( b \) is the distance from the base of the house to the bottom of the ladder, which is 6 ft,
  • \( c \) is the length of the ladder, which is 12 ft.

Substituting the known values, we have:

\[ a^2 + 6^2 = 12^2 \]

Calculating the squares:

\[ a^2 + 36 = 144 \]

Now, to solve for \( a^2 \):

\[ a^2 = 144 - 36 \] \[ a^2 = 108 \]

Next, take the square root of both sides to find \( a \):

\[ a = \sqrt{108} \] \[ a \approx 10.392 \]

Rounding to the nearest tenth, the height the ladder will reach is approximately 10.4 ft.

The correct response is:

10.4 ft.

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

62.5 ft.
62.5 ft.

4.7 ft.
4.7 ft.

22.0 ft.
22.0 ft.

7.9 ft.

To find out how far from the base of the pole each bracket should be braced into the ground, we can use the Pythagorean Theorem.

We have:

  • The length of the bracket (hypotenuse), \( c = 6.5 \) ft.
  • The height at which the bracket attaches to the pole (one leg), \( a = 4.5 \) ft.
  • The distance from the base of the pole (the other leg), \( b \) is what we need to find.

The Pythagorean Theorem states:

\[ a^2 + b^2 = c^2 \]

Substituting the known values:

\[ (4.5)^2 + b^2 = (6.5)^2 \]

Calculating the squares:

\[ 20.25 + b^2 = 42.25 \]

Now, subtract \( 20.25 \) from both sides to solve for \( b^2 \):

\[ b^2 = 42.25 - 20.25 \] \[ b^2 = 22 \]

Now, take the square root of both sides to find \( b \):

\[ b = \sqrt{22} \] \[ b \approx 4.69 \]

Rounding to the nearest tenth, the distance from the base of the pole that each bracket should be braced into the ground is approximately 4.7 ft.

The correct response is:

4.7 ft.

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 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 .

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 .

Yes, a right triangle can be formed because the Pythagorean Theorem produced a false statement.
Yes, a right triangle can be formed because the Pythagorean Theorem produced a false statement.

To determine whether a right triangle can be formed with the given sides \( a = 6 \), \( b = 10 \), and \( c = 12 \) using the Converse of the Pythagorean Theorem, we need to check if the relationship \( a^2 + b^2 = c^2 \) holds true, where \( c \) is the longest side:

  1. Calculate \( a^2 \): \[ 6^2 = 36 \]

  2. Calculate \( b^2 \): \[ 10^2 = 100 \]

  3. Calculate \( c^2 \): \[ 12^2 = 144 \]

Now, we check the relationship:

\[ 6^2 + 10^2 = 36 + 100 = 136 \] \[ 12^2 = 144 \]

Since \( 36 + 100 \neq 144 \), this means:

\[ 6^2 + 10^2 \neq 12^2 \]

Thus, according to the Converse of the Pythagorean Theorem, a right triangle cannot be formed.

The correct response is:

No, a right triangle cannot be formed because \( 6^2 + 10^2 \neq 12^2 \).