Question

Question
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 △AYB∼△XYZ
, and AB=5
, AY=12
, AY=AX
, and BY=BZ
, then what is the length of BZ
? Enter your response as a mixed fraction if needed.

(1 point)
BZ=

Answer 1

Given that triangles \( \triangle A Y B \sim \triangle X Y Z \), we can use the properties of similar triangles to find the length of \( BZ \).

From the similarity of triangles, we have:

\[
\frac{AY}{AX} = \frac{AB}{YZ}
\]

Since \( AY = AX \), let's denote \( AY = AX = 12 \).

Now, substituting the known values:

\[
\frac{12}{12} = \frac{5}{YZ}
\]

This simplification gives us:

\[
1 = \frac{5}{YZ}
\]

Thus, we find:

\[
YZ = 5
\]

Next, we are also given that \( BY = BZ \). Let us denote \( BY = BZ = x \).

Since point \( Z \) lies on the line segment \( YZ \), we can express the length of \( YZ \):

\[
YZ = YB + BZ
\]

Substituting the values we have:

\[
YZ = x + x = 2x
\]

We know from above that \( YZ = 5 \), hence:

\[
2x = 5
\]

Solving for \( x \):

\[
x = \frac{5}{2} = 2.5
\]

Thus, the length of \( BZ \) is:

\[
\boxed{2 \frac{1}{2}}
\]