Question

integrated 2
similarity quiz
name:
per.

1. each pair of figures are similar. use this fact to solve for the missing side. show all work.

a.
b.

Explanation:

Step1: Set up the proportion for similar triangles

Since \(\triangle ADE\sim\triangle ABC\) (by AA similarity, as \(\angle A\) is common and \(\angle ADE=\angle ABC\) if \(DE\parallel BC\)), the ratios of corresponding sides are equal. So, \(\frac{AD}{AB}=\frac{AE}{AC}\). Let \(AD = x\), \(AB=x + 7\), \(AE\) is not needed here. Also, using the ratio of \(DE\) - related sides (assuming \(DE\parallel BC\)): \(\frac{AD}{AB}=\frac{AE}{AC}\). Another way, if we consider the ratio of \(CE\) and \(BE\) (but more straightforward is \(\frac{x}{x + 7}=\frac{2}{8}\))

Cross - multiply: \(8x=2(x + 7)\)

Step2: Solve the equation

Expand: \(8x=2x+14\)

Subtract \(2x\) from both sides: \(8x-2x=2x + 14-2x\)

\(6x=14\)

\(x=\frac{14}{6}=\frac{7}{3}\)

For part b:

Step1: Set up the proportion for similar quadrilaterals

Since the two quadrilaterals are similar, the ratios of corresponding sides are equal. \(\frac{9}{4.5}=\frac{15}{x}\)

Step2: Solve the proportion

Cross - multiply: \(9x=15\times4.5\)

\(9x = 67.5\)

Divide both sides by \(9\): \(x=\frac{67.5}{9}=7.5\)

Answer:

a. \(x = \frac{7}{3}\)
b. \(x = 7.5\)