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3.7 ft, 14.3 ft, X

A right triangle and two of its side lengths are shown in the diagram. Which measurement is closest to the value of x in feet?
A. 190.80 ft
B. 13.81 ft
• c. 17 ft
D. 14.77 ft

7 answers

To solve for \( x \) in the right triangle with one side measuring \( 3.7 \) ft and another side measuring \( 14.3 \) ft, we can use the Pythagorean theorem, which states:

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

where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides. If we're solving for \( x \), we'll need to determine whether \( x \) is the hypotenuse or one of the legs.

We'll consider two cases:

Case 1: \( x \) is the hypotenuse

If \( x \) is the hypotenuse, we would apply the theorem as follows:

\[ x^2 = (3.7)^2 + (14.3)^2 \] \[ x^2 = 13.69 + 204.49 \] \[ x^2 = 218.18 \] \[ x = \sqrt{218.18} \approx 14.77 , \text{ft} \]

Case 2: \( x \) is one of the legs

If \( x \) is one of the legs and \( 14.3 \) ft is the hypotenuse, we would have:

\[ (14.3)^2 = (3.7)^2 + x^2 \] \[ 204.49 = 13.69 + x^2 \] \[ x^2 = 204.49 - 13.69 \] \[ x^2 = 190.80 \] \[ x = \sqrt{190.80} \approx 13.81 , \text{ft} \]

Conclusion

If \( x \) is the hypotenuse, the closest measurement to \( x \) is \( 14.77 , \text{ft} \). If \( x \) is one of the legs, the closest measurement to \( x \) is \( 13.81 , \text{ft} \).

Based on the provided answer options:

  • A. 190.80 ft
  • B. 13.81 ft
  • C. 17 ft
  • D. 14.77 ft

The appropriate closest value for \( x \) is:

D. 14.77 ft (if \( x \) is the hypotenuse) B. 13.81 ft (if \( x \) is one of the legs)

The problem does not specify which role \( x \) plays explicitly, but if you are looking for the closest value in terms of reasonable right triangle dimensions, it would seem prudent to choose D. 14.77 ft if considering it as the hypotenuse.

Which answer is it

Given the context of your question where two side lengths of a right triangle are \(3.7 , \text{ft}\) and \(14.3 , \text{ft}\), and you're looking for the possible value of \(x\):

  1. If \(x\) is the hypotenuse: \[ x \approx 14.77 , \text{ft} \]

  2. If \(x\) is one of the legs: \[ x \approx 13.81 , \text{ft} \]

Since both options \(D. 14.77 , \text{ft}\) and \(B. 13.81 , \text{ft}\) are possible based on the calculations, the best answer depends on interpreting the problem correctly.

However, if we consider the typical setup where you would expect to solve for the hypotenuse in a question about right triangles, the closest measurement for \(x\) (assuming it is the hypotenuse) is:

D. 14.77 ft.

Please choose this answer for your final selection if \(x\) is intended to be the hypotenuse.

Sammi walked from her house to school and then to her friend's house. The diagram shows the locations of these three places and their distances from each other. Which measurement is closest to the shortest distance in miles from Sammi's house to her friend's house?
From Sammi’s house 2 school is
0.4 mi
And from School to friends house is
0.2 mi
A. 0.6 mi

B. 1 mi

C. 0.45 mi

D. 0.2 mi

To find the shortest distance from Sammi's house to her friend's house, we can visualize the situation:

  1. Distance from Sammi's house to school: 0.4 miles
  2. Distance from school to friend's house: 0.2 miles

These points form a path: Sammi's house → School → Friend's house.

To find the direct distance from Sammi's house to her friend's house using the Pythagorean theorem, we can set up the situation like a right triangle, where:

  • One leg is the distance from Sammi's house to school (0.4 miles).
  • The other leg is the distance from school to her friend's house (0.2 miles).

Using the Pythagorean theorem:

\[ d = \sqrt{(0.4)^2 + (0.2)^2} \]

Calculating this:

\[ d = \sqrt{0.16 + 0.04} = \sqrt{0.20} \approx 0.447 \text{ miles} \]

Now comparing this value to the options provided:

A. 0.6 mi
B. 1 mi
C. 0.45 mi
D. 0.2 mi

The closest measurement to the direct distance (approximately 0.447 miles) is:

C. 0.45 mi.

A window washer places a 12-foot ladder 6 feet away from a house as shown in the diagram. About how high is the window from the ground?
12 ft, 6 ft
A. 10.39 ft
B. 18 ft
C. 13.42 ft
D. 6 ft

To find how high the window is from the ground, we can use the Pythagorean theorem. The setup forms a right triangle where:

  • The ladder represents the hypotenuse (\(c\)), which is 12 feet.
  • The distance from the base of the ladder to the house (\(a\)) is 6 feet.
  • The height of the window (\(b\)) from the ground is what we want to find.

According to the Pythagorean theorem:

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

Substituting the known values:

\[ (6, \text{ft})^2 + b^2 = (12, \text{ft})^2 \]

Calculating the squares:

\[ 36 + b^2 = 144 \]

Now, isolate \(b^2\):

\[ b^2 = 144 - 36 \]

\[ b^2 = 108 \]

Taking the square root to find \(b\):

\[ b = \sqrt{108} \approx 10.39 , \text{ft} \]

Now, let's look at the options provided:

A. 10.39 ft
B. 18 ft
C. 13.42 ft
D. 6 ft

Thus, the height of the window from the ground is approximately:

A. 10.39 ft.