Question

determine whether the pair of figures is similar. if so, find the scale factor. explain your reasoning.

Explanation:

Step1: Identify Angles

Both triangles \( \triangle BDC \) and \( \triangle GEC \) (assuming the other triangle is \( \triangle GEC \)) are right triangles ( \( \angle D = \angle G = 90^\circ \)) and \( \angle BCD = \angle GCE \) (vertical angles), so by AA (Angle - Angle) similarity criterion, the triangles are similar.

Step2: Find Corresponding Sides

For \( \triangle BDC \): sides are \( BD = 4 \), \( DC = 3 \), \( BC = 5 \)
For the other triangle (let's say \( \triangle GEC \)): sides are \( GE=\frac{20}{3} \), \( GC = 5 \), \( EC=\frac{25}{3} \)

Now, find the ratios of corresponding sides:

• Ratio of \( BD \) to \( GE \): \( \frac{BD}{GE}=\frac{4}{\frac{20}{3}}=\frac{4\times3}{20}=\frac{12}{20}=\frac{3}{5} \)
• Ratio of \( DC \) to \( GC \): \( \frac{DC}{GC}=\frac{3}{5} \)
• Ratio of \( BC \) to \( EC \): \( \frac{BC}{EC}=\frac{5}{\frac{25}{3}}=\frac{5\times3}{25}=\frac{15}{25}=\frac{3}{5} \)

Since all corresponding sides are in the ratio \( \frac{3}{5} \), the scale factor from \( \triangle BDC \) to the other triangle is \( \frac{3}{5} \) (or from the other triangle to \( \triangle BDC \) is \( \frac{5}{3} \), depending on the order).

Answer:

The triangles are similar by AA similarity. The scale factor (from \( \triangle BDC \) to the larger triangle) is \( \frac{5}{3} \) (or from the larger to \( \triangle BDC \) is \( \frac{3}{5} \)).