# Similarity and Proportions

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## Problem 1.

If with scale-factor 2/3, and XZ = 10, find AC. Also, if , what can you infer?

## Problem 2.

If with scale factors 2 and 3 respectively, a) Can we infer that b) If so, what is the scale-factor? c) If the perimeter of is 60, what are the perimeters of the other two triangles?

## Problem 3.

If what must be true about the triangle?

## Problem 4.

In the diagram below . Show that the scale factor is .

## Problem 5.

In the point X is the midpoint of and Y is the midpoint of . Prove that and find the scale-factor.

## Problem 6.

In the figure below, if and AC=BC then find x, y, and z. Hint: Make sure you use both of the facts that I mentioned here. Second hint: Look for similar triangles that aren't the same ones you always identify in problems that look like this.

## Problem 7.

Suppose and that is an altitude of the first triangle and is an altitude of the second. If EF=12 and BC=4 and AX=5 then find DY.

## Problem 8.

In the figure below suppose and and and . If AA''=10 and AB=15 and A''B''=5, then determine A''C.

## Problem 9.

Suppose a/b = c/d. Simplify the expression . To help you think about this, take the example where 5/4 = 25/20. In that case . From this example we could guess that, whenever a/b = c/d the expression simplifies back down to . Can you prove that this is always true?

## Problem 10.

In the figure below, suppose is equilateral. Solve for x.

## Problem 11.

In the figure below, it is given that . Prove that .

## Problem 12.

Suppose and are triangles. How much of the following information is the least amount necessary to decide whether the triangles are similar or not? I. AB=10 and DE=20 II. BC=10 and EF=20 III. CD=10 and FD=10 a) I only. b) I and II. c) III only. d) I, II, and III.