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

12 yards
12 yards

10 yards
10 yards

15 yards
15 yards

25 yards

1 answer

Given that triangles \( \triangle XYZ \sim \triangle AYB \), we can use the properties of similar triangles to establish a relationship between their corresponding sides.

We are given the following dimensions:

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

Let \( YB \) be the length we need to find. According to the property of similar triangles, the ratios of corresponding sides are equal. Therefore, we can write:

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

Substituting in the known values:

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

Now simplify \( \frac{24}{8} \):

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

Cross-multiply to solve for \( YB \):

\[ 3 \cdot YB = 30 \]

Now, divide both sides by 3:

\[ YB = \frac{30}{3} = 10 \text{ yards} \]

Thus, the length of \( YB \) is \( \boxed{10} \) yards.