Question

solve for x using the given information and the diagram below. (4 pts)$28xLXB02x=56LXB1x+2x+2x+3x+4x+6x=396LXB2x=22LXB3-1 unit$

Explanation:

First Problem (Solve for $x$ with similar figures)

Step1: Set up proportion

$\frac{2x-2}{54} = \frac{8}{18}$

Step2: Simplify right-hand side

$\frac{8}{18} = \frac{4}{9}$, so $\frac{2x-2}{54} = \frac{4}{9}$

Step3: Cross-multiply to solve

$9(2x-2) = 54 \times 4$
$18x - 18 = 216$

Step4: Isolate $x$ term

$18x = 216 + 18 = 234$

Step5: Solve for $x$

$x = \frac{234}{18} = 13$

Step1: Analyze similar triangle properties

For similar triangles: corresponding sides are proportional, corresponding angles are equal.

Step2: Evaluate each statement

I. $\frac{AB}{XY} = \frac{m\angle C}{m\angle Z}$: Ratios of sides ≠ ratios of angles (angles are equal, not proportional to sides) → False.
II. $\frac{AC}{XZ} = \frac{BC}{YZ}$: These are corresponding side ratios, which are equal for similar triangles → True.
III. $m\angle B = m\angle Y$: Corresponding angles of similar triangles are congruent → True.
IV. $AB = YZ$: Corresponding sides are proportional, not necessarily equal → False.

Step1: Define side length variables

Let sides be $x, 2x, 2x, 3x, 4x, 6x$.

Step2: Set perimeter equation

$x + 2x + 2x + 3x + 4x + 6x = 396$

Step3: Combine like terms

$18x = 396$

Step4: Solve for $x$

$x = \frac{396}{18} = 22$

Step5: Find longest side ($6x$)

$6x = 6 \times 22 = 132$

Answer:

$x=13$

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Second Problem ($\Delta ABC \sim \Delta XYZ$ true statements)