Question

Consider the similar triangles below.
Triangle 1: r-s 5, s-t 7, r-t 8.
Triangle 2: A-b _, b-c 14, a-c _.
Part A Find the scale factor. The scale factor is: __
Part B Find the length of AC. The length of AC is: _

Answer 1

To solve this problem, we first need to identify the side lengths of the two triangles based on the information given.

### Part A: Finding the Scale Factor

For Triangle 1, the side lengths are:
- \( r - s = 5 \)
- \( s - t = 7 \)
- \( r - t = 8 \)

We can check that these triangles follow the triangle inequality:
1. \( (r - s) + (s - t) = 5 + 7 = 12 > 8 \) (valid)
2. \( (r - s) + (r - t) = 5 + 8 = 13 > 7 \) (valid)
3. \( (s - t) + (r - t) = 7 + 8 = 15 > 5 \) (valid)

Now we can find the perimeter of Triangle 1:
- Perimeter = \( (r - s) + (s - t) + (r - t) = 5 + 7 + 8 = 20 \)

Now checking Triangle 2. We know:
- \( b - c = 14 \)
- Let \( A - b = x \)
- Let \( A - c = y \)

We know from the properties of similar triangles that corresponding sides are proportional. So, we can set the ratio of the sides:

\[
\frac{A - b}{r - s} = \frac{b - c}{s - t} = \frac{A - c}{r - t}
\]

Using the side lengths we have:
- \( \frac{x}{5} = \frac{14}{7} = \frac{y}{8} \)

Calculating \( \frac{14}{7} = 2 \).

Now:
\[
\frac{x}{5} = 2 \Rightarrow x = 10
\]
\[
\frac{y}{8} = 2 \Rightarrow y = 16
\]

Now, Triangle 2 has:
- \( A - b = 10 \)
- \( b - c = 14 \)
- \( A - c = 16 \)

Next, from the ratios, we can find the scale factor by choosing any corresponding side lengths:
\[
\text{Scale factor} = \frac{\text{Side in Triangle 2}}{\text{Corresponding Side in Triangle 1}} = \frac{14}{7} = 2
\]

### Part B: Finding the Length of AC

We found that \( A - c = 16 \) from our calculations:
- In Triangle 1, \( r - t = 8 \).

The length of AC in Triangle 2 is \( 16 \).

To summarize:
- **Part A: The scale factor is: 2**
- **Part B: The length of AC is: 16**