Question

Find the area of triangle whose perimeter is 48 cm and length of altitudes from opposite
vertex are 20 cm, 28 cm and 35 cm respectively

Answer 1

Find the area of triangle whose perimeter is 48 cm and length of altitudes from opposite vertex are 20 cm, 28 cm and 35 cm respectively.

Solution:

The formula of the area $\mathbf{T}$ of a triangle is:

$$\mathrm{T} = \frac{1}{2} * \mathrm{a} * \mathrm{h}_{\mathrm{a}};$$

where $\mathbf{a}$ is the length of the base of the triangle, and $\mathbf{h}_{\mathbf{a}}$ is the height of the triangle conducted to side $\mathbf{a}$.

Also we can write this formula for every side of triangle:

$$\mathrm{T} = \frac{1}{2} * \mathrm{b} * \mathrm{h}_{\mathrm{b}};$$

where $\mathbf{b}$ is the length of the base of the triangle, and $\mathbf{h}_{\mathbf{b}}$ is the height of the triangle conducted to side $\mathbf{b}$.

$$\mathrm{T} = \frac{1}{2} * \mathrm{c} * \mathrm{h}_{\mathrm{c}};$$

where $\mathbf{c}$ is the length of the base of the triangle, and $\mathbf{h}_{\mathbf{c}}$ is the height of the triangle conducted to side $\mathbf{c}$.

As you understand we can equate all this equation and we will get

$$\mathrm{T} = \frac{1}{2} * \mathrm{a} * \mathrm{h}_{\mathrm{a}} = \frac{1}{2} * \mathrm{b} * \mathrm{h}_{\mathrm{b}} = \frac{1}{2} * \mathrm{c} * \mathrm{h}_{\mathrm{c}}$$

Then

$$\mathrm{a} * \mathrm{h}_{\mathrm{a}} = \mathrm{b} * \mathrm{h}_{\mathrm{b}} = \mathrm{c} * \mathrm{h}_{\mathrm{c}}$$

The perimeter $\mathbf{P}$ of a triangle is:

$$\mathrm{P} = \mathrm{a} + \mathrm{b} + \mathrm{c}$$

In our case $\mathrm{h}_{\mathrm{a}} = 20$ cm, $\mathrm{h}_{\mathrm{b}} = 28$ cm, $\mathrm{h}_{\mathrm{c}} = 35$ cm and $\mathrm{P} = 48$ cm;

Then

$$\mathrm{a} + \mathrm{b} + \mathrm{c} = 48$$ $$20 * \mathrm{a} = 28 * \mathrm{b} = 35 * \mathrm{c}$$

Now we are solving this problem

$$\mathrm{a} = 35 * \mathrm{c} / 20$$ $$\mathrm{b} = 35 * \mathrm{c} / 28$$ $$35 * \mathrm{c} / 20 + 35 * \mathrm{c} / 28 + \mathrm{c} = 48$$ $$7 * \mathrm{c} / 4 + 5 * \mathrm{c} / 4 + \mathrm{c} = 48$$ $$\begin{array}{l} 7 * c + 5 * c + 4 * c = 192 \\ 16 * c = 192 \\ c = 12 \\ \end{array}$$

Then of the area $\mathbf{T}$ of a triangle is:

$$\mathrm{T} = \frac{1}{2} * 12 * 35 = 210$$

Answer: the area of the triangle is $210 \, \text{cm}^2$