The length of the sides of a triangle are 9 cm, 12 cm and 15 cm. Find the length of the perpendicular from the opposite vertex to the side whose length is 15 cm.

Solution

Given:
– The lengths of the sides of the triangle are 9 cm, 12 cm, and 15 cm.

We can use Heron’s formula to find the area of the triangle and then use the area to find the length of the perpendicular from the opposite vertex to the side of length 15 cm.

Step 1: Calculate the semi-perimeter (s) of the triangle

\[ s = \frac{9 + 12 + 15}{2} = 18 \text{ cm} \]

Step 2: Use Heron’s formula to find the area (A) of the triangle

\[ A = \sqrt{s(s – 9)(s – 12)(s – 15)} \]
\[ A = \sqrt{18(18 – 9)(18 – 12)(18 – 15)} \]
\[ A = \sqrt{18 \times 9 \times 6 \times 3} \]
\[ A = \sqrt{2916} \]
\[ A = 54 \text{ cm}^2 \]

Step 3: Find the length of the perpendicular (h) from the opposite vertex to the side of length 15 cm

Using the formula for the area of a triangle (\(A = \frac{1}{2} \times \text{base} \times \text{height}\)):
\[ 54 = \frac{1}{2} \times 15 \times h \]
\[ h = \frac{54 \times 2}{15} \]
\[ h = \frac{108}{15} \]
\[ h = 7.2 \text{ cm} \]

Conclusion

The length of the perpendicular from the opposite vertex to the side whose length is 15 cm is 7.2 cm.