Question

geometry exam - unit 8 name:

1. find the value of x for the similar polygons.
2. prove that the figures are similar. list the combination of transformations that occurred.
3. determine if the triangles are similar. explain your reasoning.
4. find the measure if lm. write a similarity statement.
5. determine if the triangles are similar. explain your reasoning.
6. find the value of x and write a similarity statement.

Explanation:

Problem 1

Step1: Set up proportion of sides

$\frac{2}{3} = \frac{x+3}{2x+2}$

Step2: Cross-multiply to solve for x

$2(2x+2) = 3(x+3)$
$4x + 4 = 3x + 9$

Step3: Isolate x term

$4x - 3x = 9 - 4$

1. Prove similarity: The two rectangles have all right angles (congruent corresponding angles) and their side lengths are in a consistent ratio (the larger rectangle has sides twice as long as the smaller one, so corresponding sides are proportional). By the definition of similar polygons (all corresponding angles congruent, all corresponding sides proportional), the figures are similar.
2. Transformations: First, perform a dilation of the smaller rectangle with a scale factor of 2 centered at the origin. Then, translate the dilated rectangle up and right to match the position of the larger rectangle. (Alternatively, translate first then dilate; order does not affect similarity here.)

First, calculate the missing angle in $\triangle STR$: $180^\circ - 42^\circ - 98^\circ = 40^\circ$. In $\triangle UVW$, the angles are $40^\circ$, $98^\circ$, and $180^\circ - 40^\circ - 98^\circ = 42^\circ$. All three corresponding angles of the two triangles are congruent.

Answer:

$x=5$

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Problem 2