Use the image to answer the question.

Triangle upper Y upper X upper Z is divided into triangle upper Y upper B upper A and quadrilateral upper A upper B upper Z upper X by horizontal line upper A upper B.

If △XYZ∼△AYB
, and XY¯¯¯¯¯¯¯¯=24
yards, YZ¯¯¯¯¯¯¯=30
yards, and YA¯¯¯¯¯¯¯¯=8
yards, then what is the length of YB¯¯¯¯¯¯¯¯
?

(1 point)
Responses

15 yards
15 yards

10 yards
10 yards

25 yards
25 yards

12 yards

1 answer

Given that triangles \( \triangle XYZ \sim \triangle AYB \), we can use the property of similar triangles that states the ratios of corresponding sides are equal.

Let's denote the lengths of the sides:

  • \( XY = 24 \) yards (corresponds to \( YA \))
  • \( YZ = 30 \) yards (corresponds to \( YB \))
  • \( YA = 8 \) yards (corresponds to \( YB \))

From the similarity of the triangles, we can set up the ratio:

\[ \frac{XY}{YA} = \frac{YZ}{YB} \]

Substituting the known values:

\[ \frac{24}{8} = \frac{30}{YB} \]

Calculating \( \frac{24}{8} \):

\[ 3 = \frac{30}{YB} \]

Now, we can cross-multiply to solve for \( YB \):

\[ 3YB = 30 \]

Dividing both sides by 3:

\[ YB = 10 \text{ yards} \]

Thus, the length of \( YB \) is \( 10 \) yards. The correct response is:

10 yards.