Question

Proving Similar Triangles Practice
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Question
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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.

Answer 1

To find the length of \( BZ \), we can use the property of similar triangles.

Since \( \triangle AYB \sim \triangle XYZ \), we can set up a proportion based on the corresponding sides. The given information is as follows:

- \( AB = 5 \)
- \( AY = 12 \)
- \( AX = AY = 12 \) (from the problem statement)
- \( BY = BZ \)

Let's denote the lengths:

- The side lengths of \( \triangle AYB \):
- \( AY = 12 \)
- \( AB = 5 \)

- The side lengths of \( \triangle XYZ \):
- Corresponding side \( AX = 12 \)
- Let \( BZ = x \) (which is equal to \( BY \))

Now, we can set up a proportion based on the lengths of the similar triangles:

\[
\frac{AY}{AX} = \frac{AB}{XZ} = \frac{BY}{BZ}
\]

Substituting the known values into this proportion:

\[
\frac{12}{12} = \frac{5}{XZ} = \frac{BY}{x}
\]

Since \( \frac{12}{12} = 1 \), we also have:

\[
\frac{5}{XZ} = 1 \quad \text{and therefore} \quad XZ = 5
\]

Now, since \( BY = BZ = x \) and substituting back into our proportion:

\[
\frac{BY}{BZ} = 1 \implies BY = BZ
\]

Now we can use what we have from \( BY \):

\[
\frac{12}{12} = 1 \quad \text{therefore, } BY \text{ and } BZ \text{ are equal.}
\]

From the properties of similar triangles, we know that:

\[
\frac{AB}{AY} = \frac{BY}{BZ} = \frac{5}{12}
\]

Utilizing the property \( BY = BZ \), we can say:

If \( AB = 5 \) and \( BY = x \), we can relate this through the ratios from the triangles. Setting this equal gives us:

\[
x = BY = \frac{5}{1} \cdot \frac{1}{12} = \frac{5}{12}
\]

And we already defined \( BY = BZ \):

Thus,

\[
BZ = x = \frac{5}{12}
\]

In conclusion, the length of \( BZ \) is:

\[
\boxed{\frac{5}{12}}
\]