The length of each of the two equal sides of an isosceles tria... | General Science MCQ | Examoogle

General SciencePrevious Year Paper 2024

The length of each of the two equal sides of an isosceles triangle is 89 cm each and the length of its base is 78 cm. The area (in cm 2 ) of the triangle is:

3120

3134

3102

3139

Correct Answer

🔬 SC • Science Concept & PrincipleGeneral SciencePrevious Year Paper 2024

Option A

3120

Quick Summary:

To find the area of an isosceles triangle with two equal sides of $a = 89\text{ cm}$ and a base of $b = 78\text{ cm}$ , we can use the standard area formula derived from Heron's formula or by dropping an altitude to form two congruent right-angled triangles. The altitude bisects the base into two equal segments of $\frac{78}{2} = 39\text{ cm}$ . Using the Pythagorean theorem, the height $h$ is calculated as $\sqrt{89^2 - 39^2} = \sqrt{(89-39)(89+39)} = \sqrt{50 \times 128} = \sqrt{6400} = 80\text{ cm}$ . Finally, the area is $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 78 \times 80 = 3120\text{ cm}^2$ .

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🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Think of an isosceles roof truss of a house. If you know the rafter length and the total span width, dropping a vertical support pillar right down the middle splits the roof into two symmetric right triangles, making it easy to calculate the total triangular area of the gable wall.

🧠 Memory Mnemonic / Shortcut Aid

ABH: Area equals Base times Height over two, or use Difference of Squares for quick mental math ( $a^2 - b^2 = (a-b)(a+b)$ ).

🔢 Key Formulas

$\text{Area} = \frac{1}{2} \times b \times \sqrt{a^2 - \left(\frac{b}{2}\right)^2}$ — Area of an isosceles triangle with equal sides $a$ and base $b$

$h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2}$ — Height of an isosceles triangle

⚙️ Working Principle

An isosceles triangle possesses two sides of equal length. The perpendicular dropped from the vertex between the equal sides to the base bisects the base perpendicularly, dividing the triangle into two identical right-angled triangles. By applying Pythagoras theorem ( $a^2 = b^2 + h^2$ ), we determine the height $h$ and subsequently calculate the total area using the triangle area formula.

📌 Key Points

▸
The altitude to the base of an isosceles triangle bisects the base into two equal halves.
▸
The altitude also acts as the angle bisector of the vertex angle and the median of the base.
▸
Pythagorean triplet application: $\frac{39}{39} = \frac{39}{1} \times 1$ ? No, scaling factor: $39 = 39$ , $80$ , $89$ forms a Pythagorean triple when scaled by $1$ ( $39^2 + 80^2 = 1521 + 6400 = 7921 = 89^2$ ).

✅ Advantages

▸
Provides a direct geometric route without needing full Heron's formula calculations.

❌ Disadvantages / Limitations

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Requires recognizing square numbers or factorization to simplify radicals quickly.

🛠️ Applications / Uses

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Roof truss design in civil engineering
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Land surveying and triangulation problems in competitive exams

📄 Additional Information

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Option A ( $3120$ ) is the exact correct value obtained from $\frac{1}{2} \times 78 \times 80$ .
▸
Options B, C, and D are distractor values generated by minor arithmetic errors during square root extraction or base bisection.

📊 Diagram / Illustration

✅

A is correct — The area of the isosceles triangle calculated via base and perpendicular height is $3120\text{ cm}^2$ .

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