Question

which side in △xgr corresponds with side rz?
which side in △rpz corresponds with side gr?
what is the scale factor applied to △rpz to achieve △rgx?
scale factor = (fraction)
image of two triangles with vertices p, z, r and x, g, r, and side lengths: pr=12, rz=76, xr=19, rg=3

Explanation:

Step1: Identify similar triangles

Triangles \( \triangle RPZ\) and \( \triangle RGX\) are similar. (Vertical angles are equal and alternate - interior angles are equal, so by AA (Angle - Angle) similarity criterion)

Step2: Find corresponding sides

For the side correspondence:

• In \( \triangle XGR\) and \( \triangle RPZ\), \(RZ\) corresponds to \(GR\) (because of the position of the sides with respect to the equal angles in the similar triangles).
• In \( \triangle RPZ\), \(RP\) corresponds to \(GR\) in \( \triangle XGR\) (using the similarity of triangles and the order of vertices \( \triangle RPZ\sim\triangle RGX\))

Step3: Calculate the scale factor

The scale factor \(k\) from \( \triangle RPZ\) to \( \triangle RGX\) is given by the ratio of corresponding sides. Let's take the sides \(RP = 12\) and \(RX=19\), \(RZ = 76\) and \(RG = 3\). The scale factor \(k=\frac{RX}{RP}=\frac{RG}{RZ}\) (using the property of similar triangles \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\)).

$$k=\frac{3}{76}$$

Answer:

Which side in \( \triangle XGR\) corresponds with side \(RZ\)? \(GR\)
Which side in \( \triangle RPZ\) corresponds with side \(GR\)? \(RP\)
Scale Factor \(=\frac{3}{76}\)