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Two triangular plots of land MNO and XYZ are such that the length of side MN = OM and the length of the side XY = XZ and also the angle formed at corner M is equal to the angle formed at X. The ratio of the areas of the plots MNO and XYZ in the ratio 9 ∶ 36. Find the ratio of their corresponding heights. 
1. 2 ∶ 5
2. 3 ∶ 5
3. 1 ∶ 2
4. 5 ∶ 3

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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.

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