Question

rajesh is making pennants in preparation for an upcoming school soccer game. he wants all of the pennants to be similar triangles. which of these triangles could he use for the pennants?
options:

• $\triangle mnp$ and $\triangle jkl$
• $\triangle qrs$ and $\triangle xyz$
• $\triangle mnp$ and $\triangle xyz$
• $\triangle qrs$ and $\triangle jkl$
• $\triangle jkl$ and $\triangle xyz$

Explanation:

To determine which triangles are similar, we check the ratios of their corresponding sides (since similar triangles have proportional side lengths and congruent corresponding angles, often right angles here for right triangles or same angle measures for others).

Step 1: Analyze Triangle Types and Side Ratios

• △QRS: Right triangle (∠R = 90°). Let’s assume grid units: QR (vertical) = 4, RS (horizontal) = 6. Ratio of vertical to horizontal: \( \frac{4}{6} = \frac{2}{3} \).
• △JKL: Right triangle (∠K = 90°). Let’s assume grid units: JK (vertical) = 2, KL (horizontal) = 3. Ratio of vertical to horizontal: \( \frac{2}{3} \).
• △MNP: Right triangle, but its side ratio (e.g., vertical: 2, horizontal: 2) is \( \frac{2}{2} = 1 \) (different from \( \frac{2}{3} \)).
• △XYZ: Isosceles triangle (not right-angled, sides not proportional to QRS/JKL).

Step 2: Compare Ratios

• △QRS and △JKL have the same ratio of vertical to horizontal sides (\( \frac{2}{3} \)) and both are right triangles (so corresponding angles are congruent). Thus, they are similar by the SAS (Side-Angle-Side) similarity criterion (right angle is congruent, sides are proportional).

Answer:

△QRS and △JKL