Question

practice geometry exam - unit 8name:1. find the value of x for the similar polygons.2. determine if the figures are similar and the combination of transformations that occur.3. determine if the triangles are similar. explain your reasoning.4. find the measure of ed & 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:

Step1: Set up proportion for similar rectangles

$\frac{8x-1}{12} = \frac{3x-1}{4}$

Step2: Cross-multiply to eliminate denominators

$4(8x-1) = 12(3x-1)$

Step3: Expand both sides of the equation

$32x - 4 = 36x - 12$

Step4: Rearrange terms to solve for x

$-4x = -8$
$x = \frac{-8}{-4} = 2$

The figures are similar. First, the smaller figure undergoes a translation (shift) to align with the larger figure's position. Then, it is dilated (enlarged) by a scale factor to match the size of the larger figure; the grid shows the larger figure is twice the size of the smaller one.

Calculate the third angle of the first triangle: $180^\circ - 87^\circ - 41^\circ = 52^\circ$. The second triangle has angles $52^\circ$, $41^\circ$, and $87^\circ$ (since $180^\circ - 52^\circ - 41^\circ = 87^\circ$). Both triangles have identical corresponding angles.

Step1: Set up proportion for similar triangles

$\triangle ABC \sim \triangle EBD$, so $\frac{x+1}{x+5} = \frac{12}{15}$

Step2: Simplify the right-hand fraction

$\frac{x+1}{x+5} = \frac{4}{5}$

Step3: Cross-multiply to solve for x

$5(x+1) = 4(x+5)$
$5x + 5 = 4x + 20$
$x = 15$

Step4: Calculate length of ED

$ED = x+5 = 15+5 = 20$

Answer:

$x=2$

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