Which of the following statements is NOT true?

Correct option is D

Solution:

(a) If the sides of a triangle are in the ratio 13:5:12, then two of its angles are acute angles.

This statement is TRUE. A triangle with sides in the ratio 13:5:12 is a right-angled triangle (Pythagorean triplet). In a right-angled triangle, two angles are acute, and one angle is 90°.

(b) The triangle formed by connecting the midpoints of the sides of a triangle has one-fourth area of the bigger triangle.

This statement is TRUE. The triangle formed by connecting the midpoints of a triangle’s sides is called the medial triangle, and its area is one-fourth of the original triangle's area.

(c) The area of the triangle with the largest side l always has an area less than l²/2.

This statement is TRUE. The maximum area of a triangle for a given side l occurs when the triangle is a right triangle. In this case, the area will always be less than l²/2 for non-right triangles.

(d) The area of an isosceles triangle with sides 2a, 2a, and 3a is (3√3 a²)/2.

Sides of the triangle are 2a, 2a, and 3a
​s = (2a + 2a + 3a) / 2
=> s = 7a / 2
Using Heron's formula:
​$$\begin{aligned}\text{Area} &= \sqrt{s \times (s - 2a) \times (s - 2a) \times (s - 3a)} \\\\&= \sqrt{\left(\frac{7a}{2}\right) \times \left(\frac{7a}{2} - 2a\right) \times \left(\frac{7a}{2} - 2a\right) \times \left(\frac{7a}{2} - 3a\right)} \\\\&= \sqrt{\left(\frac{7a}{2}\right) \times \left(\frac{3a}{2}\right) \times \left(\frac{3a}{2}\right) \times \left(\frac{a}{2}\right)} \\\\&= \sqrt{\left(\frac{7a}{2} \cdot \frac{3a}{2} \cdot \frac{3a}{2} \cdot \frac{a}{2}\right)} \\\\&= \sqrt{\frac{7 \cdot 3 \cdot 3 \cdot a^4}{16}} \\\\&= \frac{3\sqrt{7} \cdot a^2}{4}\end{aligned}$$​​
The given area in option 4 is (3√3 × a²) / 2, which is incorrect.


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