Question

11 create and solve two proportions to find the values of x and y. then find the measures of the indicated segments:

(\triangle abc sim \triangle ade)

(x = 10)

(y = 7.5)

(overline{ce} = 6)

(overline{bd} = 4.5)

Explanation:

Step1: Set up proportion for \(x\)

Since \(\triangle ABC\sim\triangle ADE\), the ratios of corresponding sides are equal. For the sides \(AC\) and \(AE\), \(BC\) and \(DE\), we have \(\frac{AC}{AE}=\frac{BC}{DE}\). Substituting \(AC = 4\), \(BC=5\), \(DE = 12.5\), and \(AE=x\), we get \(\frac{4}{x}=\frac{5}{12.5}\).
Cross - multiply: \(5x=4\times12.5\).

Step2: Solve for \(x\)

\(5x = 50\), then \(x=\frac{50}{5}=10\).

Step3: Set up proportion for \(y\)

Using the ratio of \(AB\) and \(AD\), \(BC\) and \(DE\). Let \(AB = 3\), \(AD=y\), \(BC = 5\), \(DE=12.5\). The proportion is \(\frac{AB}{AD}=\frac{BC}{DE}\), so \(\frac{3}{y}=\frac{5}{12.5}\).
Cross - multiply: \(5y=3\times12.5\).

Step4: Solve for \(y\)

\(5y = 37.5\), then \(y=\frac{37.5}{5}=7.5\).

Step5: Find \(CE\)

Since \(AE=x = 10\) and \(AC = 4\), then \(CE=AE - AC\). So \(CE=10 - 4=6\).

Step6: Find \(BD\)

Since \(AD=y = 7.5\) and \(AB = 3\), then \(BD=AD - AB\). So \(BD=7.5 - 3=4.5\).

Answer:

\(x = 10\), \(y = 7.5\), \(\overline{CE}=6\), \(\overline{BD}=4.5\)