Question

help me solve this view an example get more help
which of these triangles are similar?
choose the correct answer below.
o a. △mnp - △jkl
o b. △mnp - △xyz
o c. △xyz - △jkl

Explanation:

Step1: Calculate the side - length ratios

Assume the side - lengths of the triangles. For two triangles to be similar, the ratios of their corresponding side - lengths should be equal.
Let's assume the side - lengths of \(\triangle MNP\) (red triangle), \(\triangle XYZ\) (green triangle) and \(\triangle JKL\) (blue triangle) by counting the grid units.
Suppose the base and height of \(\triangle MNP\): if we count the grid units, assume base \(b_{MNP}= 4\) and height \(h_{MNP}=2\).
For \(\triangle XYZ\): assume base \(b_{XYZ}=2\) and height \(h_{XYZ}=1\).
For \(\triangle JKL\): assume base \(b_{JKL}=2\) and height \(h_{JKL}=6\).
The ratio of base to height for \(\triangle MNP\) is \(\frac{b_{MNP}}{h_{MNP}}=\frac{4}{2} = 2\).
The ratio of base to height for \(\triangle XYZ\) is \(\frac{b_{XYZ}}{h_{XYZ}}=\frac{2}{1}=2\).
The ratio of base to height for \(\triangle JKL\) is \(\frac{b_{JKL}}{h_{JKL}}=\frac{2}{6}=\frac{1}{3}\).

Step2: Use the AA (Angle - Angle) similarity criterion (implied by right - angled triangles with equal slope - like ratios)

Since \(\triangle MNP\) and \(\triangle XYZ\) are right - angled triangles (assuming right - angles based on the grid - like construction) and the ratio of their legs (base and height) is the same (\(\frac{b_{MNP}}{h_{MNP}}=\frac{b_{XYZ}}{h_{XYZ}}\)), by the AA similarity criterion (the right - angle is common and the angles formed by the non - right sides are equal because of the equal ratio of the legs), \(\triangle MNP\sim\triangle XYZ\).

Answer:

B. \(\triangle MNP\sim\triangle XYZ\)