The sides of a triangle are a=10 cm, b=14 cm & c=18 cm. A point divides the longest side in ratio 2:1. Find the area of two sub-triangles using Heron's formula.

Steps to Solve

1. Identify the sides and the ratios Given the sides of the triangle:

• $a = 10 \text{ cm}$,
• $b = 14 \text{ cm}$,
• $c = 18 \text{ cm}$. Since $c = 18 \text{ cm}$ is the longest side, the point divides it in a ratio of $2:1$.

2. Calculate the length of sub-segments The point divides the side $c$ into two segments:

• Let segment 1 (longer) be $ \frac{2}{3} \times c = \frac{2}{3} \times 18 = 12 \text{ cm}$
• Let segment 2 (shorter) be $ \frac{1}{3} \times c = \frac{1}{3} \times 18 = 6 \text{ cm}$

3. Apply Heron's Formula to the first sub-triangle For the first triangle (with sides $a$, $b$, and the longer segment of $c$):

• Calculate the semi-perimeter $s_1$: $$ s_1 = \frac{a + b + 12}{2} = \frac{10 + 14 + 12}{2} = 18 \text{ cm} $$
• Now, apply Heron’s formula: $$ A_1 = \sqrt{s_1(s_1 - a)(s_1 - b)(s_1 - 12)} $$ $$ A_1 = \sqrt{18(18-10)(18-14)(18-12)} = \sqrt{18 \times 8 \times 4 \times 6} $$

4. Calculate the area of the first sub-triangle Calculate inside the square root: $$ A_1 = \sqrt{18 \times 8 \times 4 \times 6} = \sqrt{3456} = 12\sqrt{24} \approx 41.57 \text{ cm}^2 $$

5. Apply Heron's Formula to the second sub-triangle For the second triangle (with sides $a$, segment 2 of $c$, and segment 12):

• Calculate the semi-perimeter $s_2$: $$ s_2 = \frac{a + 6 + 12}{2} = \frac{10 + 6 + 12}{2} = 14 \text{ cm} $$
• Apply Heron's formula: $$ A_2 = \sqrt{s_2(s_2 - a)(s_2 - 6)(s_2 - 12)} $$ $$ A_2 = \sqrt{14(14 - 10)(14 - 6)(14 - 12)} = \sqrt{14 \times 4 \times 8 \times 2} $$

6. Calculate the area of the second sub-triangle Calculate inside the square root: $$ A_2 = \sqrt{14 \times 4 \times 8 \times 2} = \sqrt{448} = 8\sqrt{7} \approx 21.17 \text{ cm}^2 $$