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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.(1 point)

2. Find the unknown side length in the given triangle. Round to the nearest hundredth.(1 point)
The image shows a right triangle. One of the shorter sides (let's call it the height) is labeled as 20, and the hypotenuse, which is the longest side, is labeled as 30. The base (another shorter side) of the triangle isn’t labeled in the image.

3. If a right triangle has legs measuring 10 cm and 24 cm, how long is the hypotenuse?(1 point)

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?(1 point)

5. Use the image to answer the question.A rectangle is formed from two right triangles. The base of the rectangle measures 80 yards and is labeled Johnson Avenue. The top of the rectangle is labeled Edison Avenue. The left side joining both the avenues is labeled 41st street and measures 60 yards. The right side of the rectangle is labeled 42nd street. A bank is located at the vertex of 41st street and Edison Avenue. The hypotenuse joining the bank and the vertex of Johnson Avenue and 42nd street is labeled park.Sylvia likes to walk to work when the weather is nice. She works at the bank at the corner of 41st Street and Edison Avenue. Her house is at the corner of 42nd Street and Johnson Avenue. She can walk down Johnson Avenue and turn right on 41st Street, or she can walk through the park. The distance from 42nd Street to 41st Street is 80 yards, and it is 60 yards from Johnson Avenue to Edison Avenue. Assuming the street intersections are 90° angles, how much shorter is it for Sylvia to walk through the park?(1 point)If Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be yards shorter.

3 answers

To solve the problems above, we can use the Pythagorean theorem, which states that in a right triangle:

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

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

1. Finding the other leg of the right triangle:

Given:

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

We can find the other leg \( b \):

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

\[ 144 = 36 + b^2 \]

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

\[ b = \sqrt{108} \approx 10.4 \text{ ft} \]

2. Finding the unknown side length in the second triangle:

Given:

  • Height (one leg) \( a = 20 \)
  • Hypotenuse \( c = 30 \)

We can find the base (the other leg) \( b \):

\[ 30^2 = 20^2 + b^2 \]

\[ 900 = 400 + b^2 \]

\[ b^2 = 900 - 400 = 500 \]

\[ b = \sqrt{500} \approx 22.36 \]

Rounded to the nearest hundredth, \( b \approx 22.36 \).

3. Finding the hypotenuse of another right triangle:

Given:

  • Leg \( a = 10 \) cm
  • Leg \( b = 24 \) cm

We can find the hypotenuse \( c \):

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

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

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

4. Finding the distance from the base of the house for the ladder:

Given:

  • Height of the window (one leg) \( a = 16 \) ft
  • Length of the ladder (hypotenuse) \( c = 20 \) ft

We find the distance from the wall (the other leg) \( b \):

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

\[ 400 = 256 + b^2 \]

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

\[ b = \sqrt{144} = 12 \text{ ft} \]

5. Finding how much shorter Sylvia's walk through the park is:

  • Direct path down Johnson Avenue and right on 41st Street would total:

    \( 80 \text{ yards} + 60 \text{ yards} = 140 \text{ yards} \)

  • The distance through the park using the Pythagorean theorem, with base \( 80 \) yards and height \( 60 \) yards:

\[ c^2 = 80^2 + 60^2 \]

\[ c^2 = 6400 + 3600 = 10000 \]

\[ c = \sqrt{10000} = 100 \text{ yards} \]

Difference in distance:

( 140 \text{ yards} - 100 \text{ yards} = 40 \text{ yards} )

Final answer for how much shorter it would be: If Sylvia walks through the park instead of straight down Johnson Avenue and straight down 41st Street, the walk will be 40 yards shorter.

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)Responses14.714.723423421621615.3

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

16.97
16.97

17.0
17.0

Not enough information is given to solve the problem.

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

13.4 ft.
13.4 ft.

108 ft.
108 ft.

180 ft.
180 ft.

10.4 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.

500 ft.
500 ft.

300 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

7.9 ft.
7.9 ft.

4.7 ft.
4.7 ft.

62.5 ft.
62.5 ft.

22.0 ft.

Let’s solve each problem step by step:

1. Finding the unknown side length of the triangle

Given:

  • Base \( a = 3 \)
  • Hypotenuse \( c = 15 \)
  • Perpendicular side \( b = ? \)

We can apply the Pythagorean theorem:

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

Substituting the known values:

\[ 15^2 = 3^2 + b^2 \] \[ 225 = 9 + b^2 \] \[ b^2 = 225 - 9 = 216 \] \[ b = \sqrt{216} \approx 14.7 \]

Rounding to the nearest tenth, the unknown side length \( b \) is 14.7.

2. Finding the hypotenuse of a triangle with equal legs

Given:

  • Both legs \( a = b = 12 \) cm

Using the Pythagorean theorem:

\[ c^2 = a^2 + b^2 \] \[ c^2 = 12^2 + 12^2 \] \[ c^2 = 144 + 144 = 288 \] \[ c = \sqrt{288} \approx 16.97 \]

Rounding to the nearest hundredth, the hypotenuse \( c \) is 16.97.

3. Finding how high the ladder will reach

Given:

  • Length of the ladder (hypotenuse) \( c = 12 \) ft
  • Distance from the base of the house \( a = 6 \) ft

Using the Pythagorean theorem:

\[ c^2 = a^2 + b^2 \] \[ 12^2 = 6^2 + b^2 \] \[ 144 = 36 + b^2 \] \[ b^2 = 144 - 36 = 108 \] \[ b = \sqrt{108} \approx 10.4 \]

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

4. Finding the length of the cable for the electric pole

Given:

  • Height where the cable is attached \( h = 20 \) ft
  • Distance from the cable's anchor to the base of the pole \( a = 10 \) ft

Using the Pythagorean theorem:

\[ c^2 = a^2 + h^2 \] \[ c^2 = 10^2 + 20^2 \] \[ c^2 = 100 + 400 = 500 \] \[ c = \sqrt{500} \approx 22.4 \]

Rounding to the nearest tenth, the length of the cable is 22.4 ft.

5. Finding the distance from the base of the pole for the support brackets

Given:

  • Height where the bracket will attach \( h = 4.5 \) ft
  • Length of the bracket (hypotenuse) \( c = 6.5 \) ft

Using the Pythagorean theorem:

\[ c^2 = a^2 + h^2 \] \[ 6.5^2 = a^2 + 4.5^2 \] \[ 42.25 = a^2 + 20.25 \] \[ a^2 = 42.25 - 20.25 = 22 \] \[ a = \sqrt{22} \approx 4.7 \]

Rounding to the nearest tenth, the distance from the base of the pole should be 4.7 ft.

Summary of Answers

  1. 14.7
  2. 16.97
  3. 10.4 ft
  4. 22.4 ft
  5. 4.7 ft