Question

3 - 54. decide if each pair of triangles below is similar. if the triangles are similar, justify your conclusion by stating the similarity condition you used. also describe a possible sequence of transformations that would carry one onto the other. if the triangles are not similar, explain how you know. a. b. equilateral triangles c. d.

Explanation:

Part a

Step1: Check for AA similarity

In the two right - angled triangles, one of the non - right angles (\(\alpha\)) is common. Also, both have a right angle (\(90^{\circ}\)). By the AA (Angle - Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Step2: Describe transformation

A dilation (to adjust the size) followed by a translation (to move the smaller triangle onto the larger one) would carry one triangle onto the other.

Part b

Step1: Properties of equilateral triangles

All angles in an equilateral triangle are \(60^{\circ}\). Let the side lengths of the larger and smaller equilateral triangles be \(a\) and \(b\) respectively.

Step2: Check similarity ratio

The ratio of their corresponding sides will be constant (\(\frac{a}{b}\)). By the AA similarity criterion (since all angles are \(60^{\circ}\)), the triangles are similar.

Step3: Describe transformation

A dilation (with a scale factor \(\frac{a}{b}\) or \(\frac{b}{a}\)) followed by a translation can map one equilateral triangle onto the other.

Part c

Step1: Calculate side - length ratios

\(\frac{6}{15}=\frac{2}{5}\), \(\frac{8}{20}=\frac{2}{5}\), \(\frac{11}{33}=\frac{1}{3}\)

Step2: Check similarity

Since \(\frac{6}{15}=\frac{8}{20}
eq\frac{11}{33}\), the side - length ratios are not all equal. So, the triangles are not similar.

Part d

Step1: Calculate angles

For the first triangle, using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), the third angle is \(180-(60 + 50)=70^{\circ}\). For the second triangle, the third angle is \(180-(25 + 30)=125^{\circ}\)

Step2: Check similarity

Since there are no two pairs of equal angles, the triangles are not similar.

Answer:

a. Similar (AA criterion). Dilation and translation.
b. Similar (AA criterion). Dilation and translation.
c. Not similar (side - length ratios not equal).
d. Not similar (no two pairs of equal angles).