Question

In the figure given below, ABCD is a rectangle. The area of the isosceles right triangle ABE = 7 cm²; EC = 3(BE). The area of ABCD (in cm²) is:

• A. 21 cm²
• B. 28 cm²
• C. 42 cm²
• D. 56 cm²

Hint:

For isosceles right triangles, the base and height are equal, which simplifies calculations for areas.

Answer 1

The Correct Option is B

Let \( BE = x \). Since triangle ABE is an isosceles right triangle, the base \( BE = AE \). The area of the triangle is given as: \[ \text{Area of } \triangle ABE = \frac{1}{2} \times BE \times AE = 7 \, \text{cm}^2 \] Thus: \[ \frac{1}{2} \times x \times x = 7 \Rightarrow x^2 = 14 \Rightarrow x = \sqrt{14} \] Since \( EC = 3 \times BE \), we have: \[ EC = 3x = 3\sqrt{14} \] The length of \( BC = BE + EC = x + 3x = 4x = 4\sqrt{14} \). Now, since \( AB = AE = x = \sqrt{14} \), the area of rectangle ABCD is: \[ \text{Area of ABCD} = AB \times BC = \sqrt{14} \times 4\sqrt{14} = 4 \times 14 = 56 \, \text{cm}^2 \] Thus, the Correct Answer is 56 cm².