Question

triangle pqr was changed to create triangle stu
determine if triangle pqr is similar to triangle stu.
○a. triangle pqr is similar to triangle stu because dilation (x,y) to (4x,4y) maps triangle pqr to triangle stu
○b. triangle pqr is not similar to triangle stu because geometric stretch (x,y) to (0.25x,y) maps triangle pqr to triangle stu.
○c. triangle pqr is similar to triangle stu because dilation (x,y) to (0.25x,0.25y) maps triangle pqr to triangle stu
○d. triangle pqr is not similar to triangle stu because geometric stretch (x,y) to (0.25x,0.75y) maps triangle pqr to triangle stu

Explanation:

Step1: Recall the property of similar triangles

Similar triangles can be obtained by dilation (a transformation that enlarges or reduces a figure by a scale factor). A dilation has the form \((x,y)\to(kx,ky)\) where \(k\) is the scale factor. A geometric stretch (non - uniform scaling: \((x,y)\to(k_1x,k_2y)\) with \(k_1
eq k_2\)) does not preserve similarity.

Step2: Check the coordinates of corresponding vertices

Let's assume some coordinates for simplicity. Suppose in \(\triangle PQR\), if we consider a vertex \(P\) (say \((- 4,8)\)), \(Q(4,1)\), \(R(2,-8)\) (by counting grid units). In \(\triangle STU\), if the corresponding vertices after transformation:
If we use the dilation rule \((x,y)\to(0.25x,0.25y)\):
For a point \((x,y)\) in \(\triangle PQR\), after applying \((x,y)\to(0.25x,0.25y)\)
If \(x=-4,y = 8\), then \(0.25x=-1\), \(0.25y = 2\); if \(x = 4,y=1\), then \(0.25x = 1\), \(0.25y=0.25\approx1\) (approximate due to grid - based counting). A dilation \((x,y)\to(0.25x,0.25y)\) is a uniform scaling.
A geometric stretch \((x,y)\to(k_1x,k_2y)\) with \(k_1
eq k_2\) (like in options B and D) changes the shape (ratios of side lengths) and does not preserve similarity. Option A has a scale factor \(k = 4\) which would enlarge, but visually \(\triangle STU\) is smaller.

Answer:

C. Triangle PQR is similar to triangle STU because dilation \((x,y)\) to \((0.25x,0.25y)\) maps triangle PQR to triangle STU.