Question

1. triangle is shown below. (show your work

which triangle similarity criteria proves that the two triangles above are similar?
the __________ triangle similarity criteria
find the scale factor that would be used to map a copy of abc onto rst.
the scale factor is ___________

Explanation:

Step1: Identify Similarity Criterion

First, check the angles and sides. For triangle similarity, SAS (Side - Angle - Side) criterion: if two sides of one triangle are in proportion to two sides of another triangle and the included angle is equal, triangles are similar. Let's assume we have two triangles with two sides in proportion and included angle equal. So the SAS Triangle Similarity Criterion applies.

Step2: Calculate Scale Factor

To find the scale factor from \( \triangle ABC \) to \( \triangle RST \), we take the ratio of corresponding sides. Let's say the length of a side in \( \triangle ABC \) is \( L_{ABC} \) and in \( \triangle RST \) is \( L_{RST} \). The scale factor \( k=\frac{L_{RST}}{L_{ABC}} \). Suppose the sides are such that when we divide the lengths of corresponding sides of \( \triangle RST \) by \( \triangle ABC \), we get a consistent ratio. For example, if a side of \( \triangle ABC \) is \( 13.2 \) and the corresponding side of \( \triangle RST \) is \( 4.4 \), then \( k = \frac{4.4}{13.2}=\frac{1}{3}\)? Wait, no, maybe I got the correspondence wrong. Wait, actually, to map \( ABC \) onto \( RST \), we need to find \( \frac{\text{side of } RST}{\text{side of } ABC} \). Let's check the sides. If \( ABC \) has sides, say, \( 13.2, 7.8, 8.8 \) and \( RST \) has \( 4.4, 2.6, \frac{8.8}{3}\approx2.93 \)? Wait, no, let's do it properly. Let's take corresponding sides. Suppose \( AB = 7.8 \), \( RS = 2.6 \); \( BC = 8.8 \), \( ST = \frac{8.8}{3}\approx2.93 \)? No, wait, \( 7.8\div3 = 2.6 \), \( 13.2\div3 = 4.4 \), \( 8.8\div3\approx2.93 \)? Wait, no, \( 4.4\times3 = 13.2 \), \( 2.6\times3 = 7.8 \), \( 8.8\div3\) is not. Wait, maybe the sides are \( ABC \): \( 13.2, 7.8, 8.8 \) and \( RST \): \( 4.4, 2.6, \frac{8.8}{3} \)? No, that can't be. Wait, maybe the correct ratio is \( \frac{4.4}{13.2}=\frac{1}{3} \), \( \frac{2.6}{7.8}=\frac{1}{3} \), and the included angle is equal. So the scale factor is \( \frac{1}{3} \) (since we map \( ABC \) onto \( RST \), we take \( RST \) side over \( ABC \) side).

Answer:

The \(\boldsymbol{SAS}\) Triangle Similarity Criterion
The scale factor is \(\boldsymbol{\frac{1}{3}}\)