Correct Answer - Option 3 : 1 ∶ 2
Given∶
In Δ MNO and Δ XYZ, ∠ M = ∠ X, MN = OM, XY = XZ and
ar. Δ MNO/ ar. Δ XYZ = 9 ∶ 36 = 1 ∶ 4
Formula used∶
Similarity of Triangles, SAS Theorem,
If two triangles are similar, then the ratio their areas is equal to the ratio of the squares of their corresponding sides.
Calculation∶
In Δ MNO and XYZ
MN/OM = XY/XZ
∠ M = ∠ X
∴ Δ MNO ∼ Δ XYZ [SAS Similarity]
ar. Δ MNO / ar. Δ XYZ = ON2 / YZ2 = 1/4
⇒ ON/YZ = 1/2
Also,
ar. Δ MNO / ar. Δ XYZ = (1/2 × ON × MD)/(1/2 × YZ × XN)
⇒ 1/4 = (ON/YZ) × (MD/XN)
⇒ 1/4 = (1/2) × (MD/XN)
⇒ MD/XN = (1 × 2)/(4 × 1) = 2/4 = 1/2
∴ The required ratio of the heights of the given plots is 1 ∶ 2.