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