Question

The figures shown are similar. Find the lengths of​ x, y, and z.
Question content area bottom left
Part 1
The length of side x is
  
enter your response here.
​(Type an integer or a​ decimal.)
.
.
.
Question content area right
Part 1
x
y
z
12
23
23
16
32

Answers

GPT-4o mini
To find the lengths of \(x\), \(y\), and \(z\) in similar figures, we can use proportional relationships.

Given the lengths we have:

- \( x : 12 \)
- \( y : 23 \)
- \( z : 16 \)

And their respective corresponding lengths in the similar figure:

- \( 12 : 23 \)
- \( 16 : 32 \)

### Step 1: Find \(x\)

Using the fact that the figures are similar, we can set up a proportion for \(x\):

\[
\frac{x}{12} = \frac{y}{23}
\]

Substituting \(y = 23\):

\[
\frac{x}{12} = \frac{23}{23}
\]

Since the proportion can be simplified directly:

\[
\frac{x}{12} = 1
\]

Thus, \( x = 12 \).

### Step 2: Find \(y\)

Next, we can set up a proportion to find \(y\):

\[
\frac{y}{23} = \frac{16}{32}
\]

Simplifying the right side:

\[
\frac{y}{23} = \frac{1}{2}
\]

Now we cross-multiply:

\[
2y = 23 \implies y = \frac{23}{2} = 11.5
\]

### Step 3: Find \(z\)

Finally, we can find \(z\) using the last proportion:

\[
\frac{z}{16} = \frac{12}{23}
\]

Cross-multiplying gives:

\[
23z = 192 \implies z = \frac{192}{23} \approx 8.35
\]

### Summary

So the values are approximately:

- \(x = 12\)
- \(y = 11.5\)
- \(z \approx 8.35\)

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