Question
If triangle XYZ is similar to Triangle AYB, and line XY =24 yards, line YZ =30 yards, and line YA = 8 yards then what is the length of line YB?
12 yards
10 yards
15 yards
25 yards
12 yards
10 yards
15 yards
25 yards
Answers
GPT-4o mini
If triangles XYZ and AYB are similar, the corresponding sides are in proportion. We have the lengths of sides from triangle XYZ: \( XY = 24 \) yards and \( YZ = 30 \) yards. We also have \( YA = 8 \) yards.
Let's denote \( YB \) as \( x \). The similarity of the triangles gives us the following proportion:
\[
\frac{XY}{YA} = \frac{YZ}{YB}
\]
We can substitute the given lengths into the proportion:
\[
\frac{24}{8} = \frac{30}{x}
\]
Now, simplify the left side:
\[
\frac{24}{8} = 3
\]
So we have:
\[
3 = \frac{30}{x}
\]
Now we can cross-multiply to solve for \( x \):
\[
3x = 30
\]
Next, divide both sides by 3:
\[
x = 10
\]
Thus, the length of line \( YB \) is \( 10 \) yards.
Therefore, the answer is **10 yards**.
Let's denote \( YB \) as \( x \). The similarity of the triangles gives us the following proportion:
\[
\frac{XY}{YA} = \frac{YZ}{YB}
\]
We can substitute the given lengths into the proportion:
\[
\frac{24}{8} = \frac{30}{x}
\]
Now, simplify the left side:
\[
\frac{24}{8} = 3
\]
So we have:
\[
3 = \frac{30}{x}
\]
Now we can cross-multiply to solve for \( x \):
\[
3x = 30
\]
Next, divide both sides by 3:
\[
x = 10
\]
Thus, the length of line \( YB \) is \( 10 \) yards.
Therefore, the answer is **10 yards**.