Triangle ABC is similar to triangle PQR. The ratio of their co... | Mathematics MCQ | Examoogle

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Triangle ABC is similar to triangle PQR. The ratio of their corresponding altitudes is $2:3$ . If the area of triangle ABC is $40 \text{ cm}^2$ , what is the area of triangle PQR in $\text{cm}^2$ ?

120

Correct Answer

📐 MA • Math Ratio MethodMathematicsGeometry

Option C

Quick Summary:

Since the ratio of altitudes is $2:3$ , square them to get the ratio of areas: $4:9$ . Multiply $40$ by $\frac{9}{4} = 90$ .

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📐MAMath SolutionRatio Method

📋 Given

Ratio of altitudes of similar triangles ABC and PQR is 2:3, and the area of triangle ABC is 40 cm².

🔢 Formula Used

Area(△ABC)Area(△PQR)=(Altitude1Altitude2)2\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} = \left( \frac{\text{Altitude}_1}{\text{Altitude}_2} \right)^2Area(△PQR)Area(△ABC)​=(Altitude2​Altitude1​​)2

⚡ Exam Hall Shortcut / Speed Trick

Since the ratio of altitudes is $2:3$ , square them to get the ratio of areas: $4:9$ . Multiply $40$ by $\frac{9}{4} = 90$ .

⚠️ Common Student Trap / Pitfall

Students often multiply the area by the ratio of altitudes ( $2/3$ or $3/2$ ) directly instead of squaring the ratio.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Identify property of similar triangles

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding altitudes.

Area1Area2=(h1h2)2\frac{\text{Area}_1}{\text{Area}_2} = \left( \frac{h_1}{h_2} \right)^2Area2​Area1​​=(h2​h1​​)2

Substitute given values

Substitute the ratio of altitudes $2:3$ into the formula to find the ratio of areas.

Area(△ABC)Area(△PQR)=(23)2=49\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle PQR)} = \left( \frac{2}{3} \right)^2 = \frac{4}{9}Area(△PQR)Area(△ABC)​=(32​)2=94​

Calculate target area

Given $\text{Area}(\triangle ABC) = 40 \text{ cm}^2$ , set up the equation: $\frac{40}{\text{Area}(\triangle PQR)} = \frac{4}{9}$ . Solving this gives $\text{Area}(\triangle PQR) = \frac{40 \times 9}{4} = 90$ .

Area(△PQR)=90 cm2\text{Area}(\triangle PQR) = 90 \text{ cm}^2Area(△PQR)=90 cm2

✅

C is correct because the ratio of areas is the square of the ratio of altitudes, resulting in $40 \times (3/2)^2 = 90$ .

Core Concepts Used

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💡 EXAM TIP

This property applies to all corresponding linear segments of similar triangles, including medians, angle bisectors, and perimeters.

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