Question

what are the values of $x$ and $m$? (triangles are not drawn to scale.)

Explanation:

Step1: Determine similar triangles

Since \(AB\parallel ED\), \(\triangle ABC\sim\triangle EDC\) (by AA similarity, as \(\angle ABC=\angle EDC\) and \(\angle BAC = \angle DEC\) because of parallel lines).

Step2: Set up proportion for \(x\)

For similar triangles \(\triangle ABC\) and \(\triangle EDC\), the ratios of corresponding sides are equal. So, \(\frac{AB}{ED}=\frac{AC}{EC}\).
We know \(AB = 7\), \(ED=28\), \(AC=x\), and \(EC = 32\).
Substituting values: \(\frac{7}{28}=\frac{x}{32}\).
Cross - multiply: \(28x=7\times32\).
Simplify: \(x=\frac{7\times32}{28}\).
\(x = 8\).

Step3: Set up proportion for \(m\)

Using the ratio of corresponding sides \(\frac{AB}{ED}=\frac{BC}{DC}\).
We know \(AB = 7\), \(ED = 28\), \(BC=m\), and \(DC = 40\).
Substituting values: \(\frac{7}{28}=\frac{m}{40}\).
Cross - multiply: \(28m=7\times40\).
Simplify: \(m=\frac{7\times40}{28}\).
\(m = 10\).

Answer:

\(x = 8\)
\(m = 10\)