Question

Each of equal sides of an Isosceles triangle is 2 cm greater than its height. If the base of the triangle is 12 cm, then its area is :

• A. \(18 \text{ cm}^2\)
• B. \(20 \text{ cm}^2\)
• C. \(30 \text{ cm}^2\)
• D. \(48 \text{ cm}^2\)

Hint:

1. Let height = \(h\). Then equal side \(s = h+2\). 2. The height to the base (12 cm) divides it into two segments of 6 cm each. 3. A right triangle is formed with sides \(h\), 6, and hypotenuse \(s\). 4. Use Pythagoras: \(h^2 + 6^2 = s^2 \Rightarrow h^2 + 36 = (h+2)^2\). 5. Solve for \(h\): \(h^2 + 36 = h^2 + 4h + 4 \Rightarrow 32 = 4h \Rightarrow h=8\) cm. 6. Area = \(\frac{1}{2} \times \text{base} \times h = \frac{1}{2} \times 12 \times 8 = 48 \text{ cm}^2\).

Answer 1

The Correct Option is D

Concept: In an isosceles triangle, the altitude (height) from the vertex angle to the base bisects the base and forms two congruent right-angled triangles. We can use the Pythagorean theorem and the formula for the area of a triangle. Area of a triangle = \(\frac{1}{2} \times \text{base} \times \text{height}\). Step 1: Define variables and set up relationships Let \(h\) be the height of the isosceles triangle (in cm). Let \(s\) be the length of each of the equal sides (in cm). Given: "Each of equal sides of an Isosceles triangle is 2 cm greater than its height." So, \(s = h + 2\). Given: Base of the triangle = \(12 \text{ cm}\). Step 2: Use the property of the altitude in an isosceles triangle The altitude to the base of an isosceles triangle divides the base into two equal parts. Length of half the base = \(12 \text{ cm} / 2 = 6 \text{ cm}\). This altitude (\(h\)), half the base (6 cm), and one of the equal sides (\(s\)) form a right-angled triangle. Step 3: Apply the Pythagorean theorem In the right-angled triangle formed: Hypotenuse = \(s\) One leg = \(h\) Other leg = 6 cm By Pythagorean theorem: \((\text{leg}_1)^2 + (\text{leg}_2)^2 = (\text{hypotenuse})^2\) \[ h^2 + 6^2 = s^2 \] Substitute \(s = h + 2\) into this equation: \[ h^2 + 36 = (h + 2)^2 \] Step 4: Solve for the height (\(h\)) Expand \((h+2)^2\): \((h+2)^2 = h^2 + 2(h)(2) + 2^2 = h^2 + 4h + 4\). So the equation becomes: \[ h^2 + 36 = h^2 + 4h + 4 \] Subtract \(h^2\) from both sides: \[ 36 = 4h + 4 \] Subtract 4 from both sides: \[ 36 - 4 = 4h \] \[ 32 = 4h \] Divide by 4: \[ h = \frac{32}{4} = 8 \text{ cm} \] So, the height of the triangle is 8 cm. (We can also find the equal side: \(s = h + 2 = 8 + 2 = 10 \text{ cm}\). Check: \(8^2 + 6^2 = 64 + 36 = 100 = 10^2\). This is correct.) Step 5: Calculate the area of the triangle Area = \(\frac{1}{2} \times \text{base} \times \text{height}\) Base = \(12 \text{ cm}\) Height (\(h\)) = \(8 \text{ cm}\) Area = \(\frac{1}{2} \times 12 \text{ cm} \times 8 \text{ cm}\) Area = \(6 \text{ cm} \times 8 \text{ cm}\) Area = \(48 \text{ cm}^2\) This matches option (4).