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Problem 246

# The lengths of the sides of a triangle are 6,8, and 12 . The lengths of the sides of a second triangle are $$1(1 / 2), 2$$, and 3 . Are the two triangles similar?

Expert verified
The two triangles are similar because the ratios of their corresponding sides are equal: $$\frac{6}{1\frac{1}{2}} = \frac{8}{2} = \frac{12}{3} = 4$$.
See the step by step solution

## Step 1: Find the ratios of corresponding sides

Divide the length of each side of the first triangle by the corresponding side of the second triangle. Ratio 1: $$\frac{6}{1\frac{1}{2}} = \frac{6}{\frac{3}{2}}$$ Ratio 2: $$\frac{8}{2}$$ Ratio 3: $$\frac{12}{3}$$

## Step 2: Simplify the ratios

Simplify the ratios found in Step 1. Ratio 1: To simplify this ratio, we will multiply the numerator and denominator by 2 to get rid of the fraction in the denominator. $\frac{6}{\frac{3}{2}} = \frac{6 \times 2}{\frac{3}{2} \times 2} = \frac{12}{3}$ Ratio 2: This ratio can be simplified directly. $$\frac{8}{2} = 4$$ Ratio 3: This ratio is already in simplest form. $$\frac{12}{3} = 4$$

## Step 3: Compare the ratios

Now we will compare the simplified ratios. Ratio 1: $$\frac{12}{3} = 4$$ Ratio 2: $$4$$ Ratio 3: $$4$$ All three ratios are equal to 4.

## Step 4: Determine if the triangles are similar

Since all the corresponding side ratios are equal, the two triangles are indeed similar.

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