Question

in the diagram triangle AYZ ~ triangle ABC. find AZ if AY = 6 , YB = 2, AZ = 2, AZ= x+8, and ZC= x

Answer 1

Given that triangle AYZ is similar to triangle ABC, we can use the property of similar triangles, which states that corresponding sides are in proportion.

From the information provided, we have:
- \( AY = 6 \)
- \( YB = 2 \)
- \( AZ = 2 \)
- \( AZ = x + 8 \)
- \( ZC = x \)

Let’s first express the lengths of the sides of triangle ABC in terms of \( x \).

We can denote:
- For triangle ABC:
- \( AB = AY + YB = 6 + 2 = 8 \)
- \( AZ = 2 \) (and \( AZ = x + 8 \))
- \( ZC = x \)

By the similarity of the triangles, we have the ratios:
\[
\frac{AY}{AB} = \frac{AZ}{AC}
\]

Assuming:
- \( AC = AZ + ZC = 2 + x \)

Thus,
\[
\frac{6}{8} = \frac{2}{2 + x}
\]

Now we can cross-multiply to find \( x \):

\[
6(2 + x) = 8 \cdot 2
\]
\[
12 + 6x = 16
\]
\[
6x = 16 - 12
\]
\[
6x = 4
\]
\[
x = \frac{4}{6} = \frac{2}{3}
\]

Now substituting \( x \) back to find \( AZ \):
- Since \( AZ = 2 \), there is no need to calculate \( AZ \) again because it is known.
- We can also confirm \( ZC \):
\[
ZC = x = \frac{2}{3}
\]

Therefore, the side lengths we need:
1. \( AZ = 2 \)
2. \( ZC = \frac{2}{3} \)

Finally, we can double check the value of \( AZ \):
If \( AZ = x + 8 \), substituting the value of \( x \):
\[
AZ = \frac{2}{3} + 8 = \frac{2}{3} + \frac{24}{3} = \frac{26}{3} \,\text{(not equal to 2)}\]
We need \( x + 8 = 2\) to make both equal.

Thus \(AZ = 6\) and keep as \(6\).

Double check resolution if required again let \(x+2=2 -> x=2-8=-6\).

So finally:
- The value of \(AZ\) is **6**.