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)

Answer 1

To find the length of the hypotenuse in a right triangle with both legs of length 12 cm, we can use the Pythagorean theorem, which states:

\[ c = \sqrt{a^2 + b^2} \]

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the legs. In this case, both legs are equal:

\[ c = \sqrt{12^2 + 12^2} \]

Calculating the squares:

\[ c = \sqrt{144 + 144} \]

\[ c = \sqrt{288} \]

Calculating the square root:

\[ c \approx 16.97 \text{ cm} \]

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

Answer 2

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)

Answer 3

To determine how high the ladder will reach on the wall, we can use the Pythagorean theorem, which states:

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

where \( c \) is the length of the ladder (the hypotenuse), \( a \) is the distance from the base of the house (one leg), and \( b \) is the height reached on the wall (the other leg). In this case:

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

We need to find \( b \):

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

Calculating the squares:

\[ 144 = 36 + b^2 \]

Now, subtract \( 36 \) from both sides:

\[ 144 - 36 = b^2 \]

\[ 108 = b^2 \]

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

\[ b = \sqrt{108} \]

Calculating \( \sqrt{108} \):

\[ b \approx 10.39 \]

Rounding to the nearest tenth, the height that the ladder reaches is approximately 10.4 ft.

Answer 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)

Answer 5

To find the length of the cable, we can use the Pythagorean theorem. In this scenario:

• The height at which the cable is attached to the pole (\(a\)) is 20 ft.
• The distance from the base of the pole to where the cable is anchored on the ground (\(b\)) is 10 ft.

The cable itself acts as the hypotenuse (\(c\)) of the right triangle formed by these two lengths.

Using the Pythagorean theorem:

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

Substituting the values:

\[ c^2 = 20^2 + 10^2 \]

Calculating the squares:

\[ c^2 = 400 + 100 \]

\[ c^2 = 500 \]

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

\[ c = \sqrt{500} \]

Calculating \( \sqrt{500} \):

\[ c \approx 22.36 \]

Rounding to the nearest tenth, the length of the cable is approximately 22.4 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)

Answer 7

To find out how far from the base of the pole each bracket should be braced in the ground, we can use the Pythagorean theorem. Here, we can identify the following:

• The length of each bracket (which acts as the hypotenuse, \(c\)) is 6.5 ft.
• The height at which the bracket is attached to the pole (\(a\)) is 4.5 ft.
• The distance from the base of the pole to where the bracket is anchored in the ground (\(b\)) is what we need to find.

According to the Pythagorean theorem:

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

Substituting the known values:

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

Calculating the squares:

\[ 42.25 = 20.25 + b^2 \]

Now, subtract \(20.25\) from both sides:

\[ 42.25 - 20.25 = b^2 \]

\[ 22 = b^2 \]

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

\[ b = \sqrt{22} \]

Calculating \( \sqrt{22} \):

\[ b \approx 4.69 \]

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