Question

problem 30
triangle abc is dilated using center d to form triangle a b c
what is the measurement of side a b ?

Explanation:

Step1: Determine similarity

Triangles \(ABC\) and \(A'B'C'\) are similar (by AA similarity as they are right - angled and have the same shape).

Step2: Set up proportion

For similar triangles, the ratios of corresponding sides are equal. Let \(A'B' = x\). The ratio of the vertical sides is \(\frac{4}{12}\), and the ratio of the hypotenuses (we can use the ratio of vertical - to - hypotenuse, but since for similar triangles \(\frac{\text{side}_1}{\text{side}_2}=\frac{\text{corresponding side}_1}{\text{corresponding side}_2}\)). Using the ratio of the vertical sides and hypotenuses: \(\frac{3}{x}=\frac{9}{A'B}\). Wait, better using Pythagorean theorem for small triangle: \(AB=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5\), for large triangle \(A'B'=\sqrt{9^{2}+12^{2}}=\sqrt{81+144}=\sqrt{225}\)

Another approach: Since the triangles are similar, the scale factor \(k=\frac{12}{4}=3\) (ratio of vertical sides).

Step3: Calculate \(A'B'\)

If the scale factor \(k = 3\) and \(AB = 5\) (from \(AB=\sqrt{3^{2}+4^{2}}=5\)), then \(A'B'=k\times AB\)

\(A'B'=3\times5 = 15\)

Answer:

\(15\)