Question

Question

If a right triangle has a leg that is 6 ft. long and the hypotenuse is 12 ft. long, how long is the other leg? Round to the nearest tenth.

Answer 1

To find the length of the other leg of a right triangle when one leg and the hypotenuse are known, we can use the Pythagorean theorem, which states:

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

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

Given:

• One leg \( a = 6 \) ft
• Hypotenuse \( c = 12 \) ft

We need to find the length of the other leg \( b \).

Using the Pythagorean theorem:

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

Calculating the squares:

\[ 36 + b^2 = 144 \]

Now, we isolate \( b^2 \):

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

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

\[ b = \sqrt{108} \]

Now we can simplify \( \sqrt{108} \):

\[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3} \]

Next, we can approximate \( \sqrt{3} \approx 1.732 \):

\[ b \approx 6 \times 1.732 \approx 10.392 \]

Rounding to the nearest tenth:

\[ b \approx 10.4 \]

So, the length of the other leg is approximately 10.4 ft.

Answer 2

If a right triangle has legs measuring 10 cm and 24 cm, how long is the hypotenuse?

Answer 3

To find the length of the hypotenuse of a right triangle given the lengths of the two legs, we use the Pythagorean theorem, which states:

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

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

Given:

• \( a = 10 \) cm
• \( b = 24 \) cm

We can calculate the hypotenuse \( c \):

\[ c^2 = 10^2 + 24^2 \]

Calculating the squares:

\[ c^2 = 100 + 576 \] \[ c^2 = 676 \]

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

\[ c = \sqrt{676} \] \[ c = 26 \]

Thus, the length of the hypotenuse is 26 cm.

Answer 4

Camila needs to fix the window on the front of her house. The bottom of the window is 16 ft. from the ground. She only has a 20 ft. ladder. How far from the base of the house should she set up the ladder so that the top of the ladder is exactly at the bottom of the window?

Answer 5

To find out how far from the base of the house Camila should set up the ladder, we can use the Pythagorean theorem. In this scenario, the ladder acts as the hypotenuse of a right triangle, where:

• The height from the ground to the bottom of the window (one leg of the triangle) is \( 16 \) ft.
• The length of the ladder (the hypotenuse) is \( 20 \) ft.
• The distance from the base of the house to the bottom of the ladder (the other leg of the triangle) is what we are trying to find.

Let \( d \) be the distance from the base of the house to the bottom of the ladder. According to the Pythagorean theorem:

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

Here, we have:

• \( a \) (height of the window) = \( 16 \) ft
• \( b \) (distance from the base of the house to the bottom of the ladder) = \( d \)
• \( c \) (length of the ladder) = \( 20 \) ft

Substituting the values into the equation:

\[ 16^2 + d^2 = 20^2 \]

Calculating the squares:

\[ 256 + d^2 = 400 \]

Now isolate \( d^2 \):

\[ d^2 = 400 - 256 \] \[ d^2 = 144 \]

Taking the square root of both sides:

\[ d = \sqrt{144} \] \[ d = 12 \]

Therefore, Camila should set up the ladder 12 ft from the base of the house.

Answer 6

.

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

15.3
15.3

14.7
14.7

234
234

216

Answer 7

To find the length of the unknown side in the right triangle using the Pythagorean theorem, let's denote the sides as follows:

• Let \( a = 3 \) (the base)
• Let \( b \) (the unknown side that we need to find)
• Let \( c = 15 \) (the hypotenuse)

According to the Pythagorean theorem:

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

Substituting the known values into the equation:

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

Calculating the squares:

\[ 9 + b^2 = 225 \]

Now, isolate \( b^2 \):

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

Taking the square root of both sides:

\[ b = \sqrt{216} \]

Now, we can simplify \( \sqrt{216} \):

\[ \sqrt{216} = \sqrt{36 \times 6} = \sqrt{36} \times \sqrt{6} = 6\sqrt{6} \]

Next, we approximate \( \sqrt{6} \approx 2.449 \):

\[ b \approx 6 \times 2.449 \approx 14.694 \]

Rounding to the nearest tenth:

\[ b \approx 14.7 \]

Therefore, the unknown side length rounded to the nearest tenth is \( \boxed{14.7} \).

Answer 8

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

16.97
16.97

288
288

Not enough information is given to solve the problem.

Answer 9

To find the length of the hypotenuse of a right triangle when both legs are given, we can use the Pythagorean theorem, which states:

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

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

In this case:

• \( a = 12 \) cm
• \( b = 12 \) cm

Substituting these values into the equation:

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

Calculating the squares:

\[ c^2 = 144 + 144 \] \[ c^2 = 288 \]

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

\[ c = \sqrt{288} \]

Next, we can simplify \( \sqrt{288} \):

\[ \sqrt{288} = \sqrt{144 \times 2} = \sqrt{144} \times \sqrt{2} = 12\sqrt{2} \]

Now, we can approximate \( \sqrt{2} \approx 1.414 \):

\[ c \approx 12 \times 1.414 \approx 16.968 \]

Rounding to the nearest hundredth:

\[ c \approx 16.97 \]

Therefore, the length of the hypotenuse rounded to the nearest hundredth is \( \boxed{16.97} \).

Answer 10

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.

13.4 ft.
13.4 ft.

180 ft.
180 ft.

10.4 ft.

Answer 11

To determine how high the ladder will reach, we can apply the Pythagorean theorem. In this scenario:

• Let \( c \) be the length of the ladder (the hypotenuse).
• Let \( a \) be the height reached by the ladder (which we are trying to find).
• Let \( b \) be the distance from the base of the house to the bottom of the ladder.

Given:

• \( c = 12 \) ft (length of the ladder)
• \( b = 6 \) ft (distance from the base of the house)

Using the Pythagorean theorem:

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

Substituting the known values:

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

Calculating the squares:

\[ a^2 + 36 = 144 \]

Now, isolate \( a^2 \):

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

Taking the square root of both sides to find \( a \):

\[ a = \sqrt{108} \]

We can simplify \( \sqrt{108} \):

\[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3} \]

Next, we can approximate \( \sqrt{3} \approx 1.732 \):

\[ a \approx 6 \times 1.732 \approx 10.392 \]

Rounding to the nearest tenth:

\[ a \approx 10.4 \]

Therefore, the height the ladder will reach, rounded to the nearest tenth, is \( \boxed{10.4} \) ft.