Question

which triangles are similar to △a? choose all that apply. triangle d triangle b triangle e triangle c

Explanation:

Step1: Calculate side - length ratios

Let the side - length of each small square be \(1\).
For \(\triangle A\), assume the horizontal side \(a = 3\), the vertical side \(b = 2\), and the hypotenuse \(c=\sqrt{3^{2}+2^{2}}=\sqrt{13}\).
For \(\triangle B\), the horizontal side \(a_{1}=1.5\), the vertical side \(b_{1}=1\), and the hypotenuse \(c_{1}=\sqrt{1.5^{2}+1^{2}}=\sqrt{3.25}=\frac{\sqrt{13}}{2}\). Then \(\frac{a_{1}}{a}=\frac{1.5}{3}=\frac{1}{2}\), \(\frac{b_{1}}{b}=\frac{1}{2}\), \(\frac{c_{1}}{c}=\frac{1}{2}\).
For \(\triangle D\), the horizontal side \(a_{2}=3\), the vertical side \(b_{2}=2\), and the hypotenuse \(c_{2}=\sqrt{3^{2}+2^{2}}=\sqrt{13}\). Then \(\frac{a_{2}}{a}=\frac{3}{3} = 1\), \(\frac{b_{2}}{b}=\frac{2}{2}=1\), \(\frac{c_{2}}{c}=1\).

Step2: Check similarity criteria

By the Side - Side - Side (SSS) similarity criterion (\(\frac{a_{1}}{a}=\frac{b_{1}}{b}=\frac{c_{1}}{c}\) for \(\triangle A\) and \(\triangle B\), \(\frac{a_{2}}{a}=\frac{b_{2}}{b}=\frac{c_{2}}{c}\) for \(\triangle A\) and \(\triangle D\)).

Answer:

Triangle \(B\), Triangle \(D\)