Question

determine whether the following pairs of triangles, (i)-(iv), are similar. if they are similar, explain why. find the scale factor of the triangles that are similar. the figures are not drawn to scale

d. the triangles are not similar

(ii) select the correct choice below and, if necessary, fill in the answer box to complete your choice

a. \\( \triangle a b c - \triangle e d a \\) by the sss similarity for triangles theorem. the scale factor of \\( \triangle a b c \\) to \\( \triangle e d a \\) is

(type an integer or a simplified fraction.)

b. \\( \triangle a b c - \triangle e d a \\) by the aa similarity for triangles theorem. the scale factor of \\( \triangle a b c \\) to \\( \triangle e d a \\) is

(type an integer or a simplified fraction.)

c. \\( \triangle a b c - \triangle e d a \\) by the ssa similarity for triangles theorem. the scale factor of \\( \triangle a b c \\) to \\( \triangle e d a \\) is

(type an integer or a simplified fraction.)

d. the triangles are not similar

Explanation:

Step1: Check for right angles and common angles

In \(\triangle ABC\) and \(\triangle EDA\), \(\angle B=\angle D = 90^{\circ}\). Also, \(\angle BAC+\angle BCA = 90^{\circ}\) and \(\angle BCA+\angle DEA=90^{\circ}\) (since \(\angle D = 90^{\circ}\)), so \(\angle BAC=\angle DEA\) (by the property that if two angles are complementary to the same angle, they are equal).

Step2: Apply AA similarity theorem

Since two angles of \(\triangle ABC\) (\(\angle B = 90^{\circ}\) and \(\angle BAC\)) are equal to two angles of \(\triangle EDA\) (\(\angle D=90^{\circ}\) and \(\angle DEA\)), by the AA (Angle - Angle) Similarity for Triangles theorem, \(\triangle ABC\sim\triangle EDA\).

Step3: Calculate the scale factor

The scale factor of \(\triangle ABC\) to \(\triangle EDA\) is \(\frac{AB}{ED}\). Given \(AB = 7\) and \(ED=14\), the scale factor is \(\frac{7}{14}=\frac{1}{2}\)

Answer:

B. \(\triangle ABC\sim\triangle EDA\) by the AA Similarity for Triangles theorem. The scale factor of \(\triangle ABC\) to \(\triangle EDA\) is \(\frac{1}{2}\)