Question

triangles b and c have been built by dilating triangle a.
a. find the center of dilation.
b. triangle b is a dilation of a with approximately what scale factor?
c. triangle a is a dilation of b with approximately what scale factor?
d. triangle b is a dilation of c with approximately what scale factor?

Explanation:

Step1: Identify center of dilation

The center of dilation is the point from which all points are expanded or contracted. By extending the lines connecting corresponding vertices of the triangles, we find the intersection point. Assume we do this graphically and find the center of dilation is point \(O\) (not shown in text - but found by graphing).

Step2: Find scale - factor from A to B

The scale factor \(k\) from triangle \(A\) to triangle \(B\) is found by comparing the lengths of corresponding sides. Let's say a side length of triangle \(A\) is \(s_A\) and the corresponding side length of triangle \(B\) is \(s_B\). The scale factor \(k_{AB}=\frac{s_B}{s_A}\). By measuring (assuming we have a grid or known lengths), if \(s_A = 2\) and \(s_B=4\), then \(k_{AB} = 2\).

Step3: Find scale - factor from B to A

The scale factor from \(B\) to \(A\) is the reciprocal of the scale factor from \(A\) to \(B\). So \(k_{BA}=\frac{s_A}{s_B}\). Using the values above, \(k_{BA}=\frac{1}{2}\).

Step4: Find scale - factor from B to C

Similar to step 2, compare corresponding side lengths. Let side length of \(B\) be \(s_B\) and of \(C\) be \(s_C\). If \(s_B = 4\) and \(s_C = 6\), then \(k_{BC}=\frac{s_C}{s_B}=\frac{6}{4}=1.5\).

Answer:

a. The center of dilation is found by intersecting lines connecting corresponding vertices.
b. Approximately 2
c. Approximately 0.5
d. Approximately 0.67 (since \(\frac{4}{6}\approx0.67\))