Final answer:
In similar triangles Δ LMN and Δ PQR, with a given ratio of the areas 9Ar(Δ PQR) = 16 Ar(Δ LMN), and a known side length QR = 20, the length MN is found to be 15 units.
Step-by-step explanation:
The ratio of the areas of similar triangles is the square of the ratio of their corresponding sides.
( Ar(Δ PQR) / Ar(Δ LMN) = (QR/LN)²
Given that 9Ar(Δ PQR) = 16 Ar(Δ LMN), we have (QR/LN)² = 9/16.
Substitute QR = 20 into the equation.
- ( (20/LN)² = 9/16.
- Cross-multiply: 20² = 9 × 16
- Solve for LN:LN = 20 × √9/16 = 20 × 3/4 = 15.
The length MN of triangle Δ LMN is determined to be 15 units. This calculation is based on the ratio of the areas of the two similar triangles and the known side length \( QR \) in accordance with the properties of similar triangles.
Complete Question:
Δ LMN ∼ ΔPQR, 9Ar(ΔPQR) = 16 Ar(ΔLMN). If QR=20 then Find MN.