Question

question 6 of 10

what is the distance from point \\(y\\) to \\(\overleftrightarrow{wx}\\) in the figure below?

a. 5
b. \\(10\sqrt{2}\\)
c. \\(10\sqrt{3}\\)
d. cannot be determined
e. 10
f. 35

Explanation:

Identify the distance segment

The distance from a point to a line is the length of the perpendicular segment from the point to the line. In the given figure, segment \(YZ\) is perpendicular to the line \(\overleftrightarrow{WX}\) at point \(Z\). Therefore, the distance from point \(Y\) to \(\overleftrightarrow{WX}\) is the length of \(YZ\).

Set up the right triangles

The perpendicular segment \(YZ\) divides the figure into two right triangles: \(\triangle WZY\) and \(\triangle XZY\), both sharing the side \(YZ\).
We are given:

• \(WY = 26\)
• \(WZ = 24\)
• \(ZX = 10\)
• \(XY = 10\sqrt{2}\)

Apply the Pythagorean theorem

We can use either right triangle to find the length of \(YZ\). Let's use \(\triangle WZY\), which is a right triangle with hypotenuse \(WY\) and legs \(WZ\) and \(YZ\).

$$ WZ^2 + YZ^2 = WY^2 $$

Solve for the unknown length

Substitute the known values into the equation:

$$ 24^2 + YZ^2 = 26^2 $$

$$ 576 + YZ^2 = 676 $$

$$ YZ^2 = 676 - 576 = 100 $$

$$ YZ = \sqrt{100} = 10 $$

Verify with the second triangle

We can verify this result using the second right triangle, \(\triangle XZY\):

$$ ZX^2 + YZ^2 = XY^2 $$

$$ 10^2 + 10^2 = 100 + 100 = 200 $$

Since \(XY = 10\sqrt{2}\), we have:

$$ XY^2 = (10\sqrt{2})^2 = 100 \times 2 = 200 $$

The values are consistent, confirming that the distance \(YZ\) is \(10\).

Answer:

• A. 5
• B. \(10\sqrt{2}\)
• C. \(10\sqrt{3}\)
• D. Cannot be determined
• E. 10 (Correct answer)
• F. 35