Two similar triangles are $\Delta ABC $ and $\Delta LMN $. If ... | Mathematics MCQ | Examoogle

MathematicsPrevious Year Paper 2024

Two similar triangles are $\Delta ABC$ and $\Delta LMN$ . If the area of $\Delta ABC = 25 \text{ cm}^2$ , area of $\Delta LMN = 36 \text{ cm}^2$ and $BC = 2.5 \text{ cm}$ , then the measure of $MN$ (in cm) is:

Correct Answer

📐 MA • Math Area Ratio TheoremMathematicsPrevious Year Paper 2024

Option A

Quick Summary:

Take the square root of the ratio of areas first to get the side ratio: $\sqrt{25/36} = 5/6$ . Then, equate $5/6 = 2.5/MN$ , implying $MN = 2.5 \times 6 / 5 = 3$ . Wait, recalculating: $5 \times MN = 6 \times 2.5 = 15$ , so $MN = 3$ . Let me re-verify given options.

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📐MAMath SolutionArea Ratio Theorem

📋 Given

Area of $\Delta$ ABC = 25 cm², Area of $\Delta$ LMN = 36 cm², BC = 2.5 cm. Triangles are similar.

🔢 Formula Used

Area(ΔABC)Area(ΔLMN)=(BCMN)2\frac{\text{Area}(\Delta ABC)}{\text{Area}(\Delta LMN)} = \left( \frac{BC}{MN} \right)^2Area(ΔLMN)Area(ΔABC)​=(MNBC​)2

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often forget to take the square root of the area ratio before equating it to the side ratio, or they confuse the numerator and denominator.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Apply Area of Similar Triangles Theorem

The ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Area(ΔABC)Area(ΔLMN)=(BCMN)2\frac{\text{Area}(\Delta ABC)}{\text{Area}(\Delta LMN)} = \left( \frac{BC}{MN} \right)^2Area(ΔLMN)Area(ΔABC)​=(MNBC​)2

Substitute given values

Substitute the areas and the length of $BC$ into the equation.

2536=(2.5MN)2\frac{25}{36} = \left( \frac{2.5}{MN} \right)^23625​=(MN2.5​)2

Take square root on both sides

Take the square root of the entire equation to solve for the side ratio.

2536=2.5MN  ⟹  56=2.5MN\sqrt{\frac{25}{36}} = \frac{2.5}{MN} \implies \frac{5}{6} = \frac{2.5}{MN}3625​​=MN2.5​⟹65​=MN2.5​

Solve for MN

Cross-multiply to isolate $MN$ . $5 \times MN = 6 \times 2.5$ , which equals $15$ . Therefore, $MN = 15/5 = 3$ .

MN=2.5×65=3MN = \frac{2.5 \times 6}{5} = 3MN=52.5×6​=3

✅

C is correct because the calculation based on the square root of the ratio of areas $(5:6)$ applied to the side $BC=2.5$ yields $MN=3$ . Note: Although the key says D, the mathematically correct answer based on the provided values is 3.

Core Concepts Used

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💡 EXAM TIP

This concept is a staple in geometry; remember that ratios of perimeters, medians, and altitudes also follow the linear ratio, while areas follow the square ratio.

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