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Problem 269

# The sides of triangle $$\mathrm{ABC}$$ measure 5,7, and 9 . The shortest side of a similar triangle, $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$, measures 10 . (a) Find the measure of the longest side of triangle $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$. (b) Find the ratio of the measures of a pair of corresponding altitudes in triangles $$\mathrm{ABC}$$ and $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$. (c) Find the perimeter of triangle $\mathrm{A}^{\prime} \mathrm{B}^{\prime} \mathrm{C}^{\prime}$.

Expert verified
The longest side of triangle A'B'C' measures 18 units, the ratio of the measures of a pair of corresponding altitudes in triangles ABC and A'B'C' is 2, and the perimeter of triangle A'B'C' is 42 units.
See the step by step solution

## Step 1: Find the scale factor between the two triangles

Since the triangles are similar, the ratio of their corresponding sides is constant. We can find this ratio (scale factor) by dividing the given side of triangle A'B'C' (10) by the corresponding side of triangle ABC (5): Scale factor = $$\frac{10}{5}$$ = 2

## Step 2: Find the longest side of triangle A'B'C'

Now that we have the scale factor, we can find the longest side of A'B'C' by multiplying the longest side of triangle ABC (9) by the scale factor (2): Longest side of A'B'C' = 9 × 2 = 18 So, the longest side of triangle A'B'C' measures 18 units.

## Step 3: Find the ratio of the measures of a pair of corresponding altitudes

Because the triangles are similar, the ratio of their altitudes is the same as the scale factor between them. So, the ratio of the measures of a pair of corresponding altitudes is: Ratio of altitudes = Scale factor = 2 This means that if the altitude in triangle ABC is x, the corresponding altitude in triangle A'B'C' is 2x.

## Step 4: Find the perimeter of triangle A'B'C'

Now that we have the scale factor, we can find the perimeter of triangle A'B'C'. We know that the sides of triangle ABC are 5, 7, and 9. To find the corresponding sides of triangle A'B'C', we can multiply each side by the scale factor (2): Side A'B' = 10 (given) Side B'C' = 7 × 2 = 14 Side C'A' = 18 (found in step 2) Perimeter of triangle A'B'C' = 10 + 14 + 18 = 42 Thus, the perimeter of triangle A'B'C' is 42 units.

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