Question

Find the area of a right-angled triangle, the radius of whose circumcircle measures $$8\text{cm}$$, and the altitude drawn to the hypotenuses measures $$\text{6cm}$$.

Answer 1

Hint: Here, we will use the formula for finding the area of a right-angled triangle as shown below:
$$\text{Area of }\mathrm{\Delta }\text{ABC = }{\displaystyle \frac{1}{2}}\times b\text{ase}\times \text{height}$$

Complete step by step answer:
Step (1): First of all, we will draw the diagram according to the given information in the question where $$\mathrm{\Delta }\text{ABC}$$ is a right-angled triangle with a circumcircle having Centre O:

Where, $$\text{AO}=\text{BO}=\text{CO}=r=8\text{cm}$$.
Step 2: Now we will draw a perpendicular from the point $$\text{B}$$to the hypotenuse $$\text{AC}$$ which touches it at the point $$\text{D}$$ as shown below:

So, $$\text{BD = 6cm}$$ ($$\because$$ as given)
Step 3: As we know that the radius of the circle is equal so, $$\text{AO}=\text{CO}$$ and for the hypotenuse of the triangle is equals to the sum of both the radius as shown below:
$$\Rightarrow \text{AO + CO = AC}$$
By substituting the values of the radius $$\text{AO}=\text{CO}=r=8\text{cm}$$ in $$\text{AO + CO = AC}$$, we get:
$$\Rightarrow \text{AC = 16cm}$$
Step 4: Now, in the right-angled triangle $$\mathrm{\Delta }\text{ABC}$$, the altitude of the triangle is $$\text{BD = 6cm}$$ and the hypotenuse is $$\text{AC = 16cm}$$. By using the formula of area of the triangle which states that area of right-angle triangle = $${\displaystyle \frac{1}{2}}\times \text{base}\times \text{height}$$, we get:
$$\Rightarrow \text{Area of }\mathrm{\Delta }\text{ABC = }{\displaystyle \frac{1}{2}}\times 16\times 6$$ ……………. (1)
By solving the RHS side of the above expression (1), we get:
$$\Rightarrow \text{Area of }\mathrm{\Delta }\text{ABC = 48c}{\text{m}}^2$$

The area of the triangle is $$\text{48c}{\text{m}}^2$$.

Note:
Students needs to remember that the height circumcenter of a right-angle triangle is the midpoint of its hypotenuse:
$$\Rightarrow \text{Hypotenuse}=2\times \text{Radius of the circumcircle}$$
In these questions, for finding the area of the triangle, we need to consider hypotenuse as a base and the altitude to the hypotenuse as height.