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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?

the sides are 3.7 ft and 14.3 ft

9 answers

To find the value of \( x \) in a right triangle where one side is \( 3.7 \) ft and the other side is \( 14.3 \) ft, we can 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 other two sides.

Assuming \( 3.7 \) ft and \( 14.3 \) ft are the lengths of the legs, we can find \( x \) (the hypotenuse) as follows:

  1. Square each of the lengths: \[ a^2 = (3.7)^2 = 13.69 \] \[ b^2 = (14.3)^2 = 204.49 \]

  2. Add these squares: \[ c^2 = a^2 + b^2 = 13.69 + 204.49 = 218.18 \]

  3. Take the square root to find \( c \): \[ c = \sqrt{218.18} \approx 14.76 \]

Thus, the value of \( x \) (the hypotenuse) is closest to 14.8 ft.

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 friends house?

To determine the shortest distance from Sammi's house to her friend's house based on the given information and diagram (which I cannot see), we need to apply the concept of distance in a triangle, assuming Sammi walked from her house to school and then to her friend's house.

Let's say:

  • A = Sammi's house
  • B = School
  • C = Friend's house

If we consider the distances as follows:

  • AB = Distance from Sammi's house to the school
  • BC = Distance from the school to her friend's house
  • AC = Distance from Sammi's house to her friend's house (which we need to find)

To find the shortest distance (which could be the direct path from point A to C), we can use the triangle inequality or the Pythagorean theorem if the points form a right triangle.

  1. If the triangle formed (A, B, C) is a right triangle, then you can use: \[ AC^2 = AB^2 + BC^2 \] Calculate \( AC \) from the distances provided.

  2. If it is not a right triangle, then simply calculate the direct distance \( AC \) based on the coordinates (if provided) or approximations using the triangle inequality.

If you can provide the specific distances between these points or describe the layout of the triangle (right triangle, obtuse triangle, etc.), I could help you calculate the exact distance. Without the diagram or distances, I cannot provide a numerical answer.

the distance between the school and sammi's house is 0.4 mi and the distance between the school and the friends house is 0.2 mi

To find the shortest distance from Sammi's house to her friend's house, we can use the given distances:

  • Distance from Sammi's house to school (A to B) = 0.4 miles
  • Distance from school to her friend's house (B to C) = 0.2 miles

If we assume these points form a triangle, the shortest distance from Sammi's house (A) to her friend's house (C) would generally be the direct line (AC) between them, which can be found using the Pythagorean theorem if we know the angle between the two distances. However, since no angle is provided, we can draw the triangle based on the following assumptions:

  1. If the route from Sammi's house to her friend's house forms a right triangle:

    • Let \( AC \) be the hypotenuse with:
      • \( AB = 0.4 \)
      • \( BC = 0.2 \)

  2. To find \( AC \):

\[ AC^2 = AB^2 + BC^2 \] \[ AC^2 = (0.4)^2 + (0.2)^2 \] \[ AC^2 = 0.16 + 0.04 \] \[ AC^2 = 0.2 \] \[ AC = \sqrt{0.2} \approx 0.447 \text{ miles} \]

So, the shortest distance from Sammi's house to her friend's house is approximately 0.45 miles.

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?

To find out how high the window is from the ground using the information provided about the ladder, we can use the Pythagorean theorem.

Given:

  • Length of the ladder (hypotenuse) = 12 feet
  • Distance from the house (one leg of the right triangle) = 6 feet
  • Height of the window (the other leg of the right triangle) = \( h \)

The relationship can be described using the Pythagorean theorem:

\[ \text{(hypotenuse)}^2 = \text{(leg 1)}^2 + \text{(leg 2)}^2 \]

In this case:

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

Calculating:

\[ 144 = 36 + h^2 \]

Subtract \( 36 \) from both sides:

\[ 144 - 36 = h^2 \]

\[ 108 = h^2 \]

Now, take the square root of both sides:

\[ h = \sqrt{108} \approx 10.39 \text{ feet} \]

Thus, the height of the window from the ground is approximately 10.4 feet.

the answer choices are

10.39 ft

18 ft

13.42 ft

6 ft

Based on the calculations we provided earlier, the height of the window from the ground is approximately \( 10.39 \) feet.

Given the answer choices you provided:

  • 10.39 ft
  • 18 ft
  • 13.42 ft
  • 6 ft

The correct choice is 10.39 ft.