Question

Find the maximum area of the triangle inscribed in a semi-circle having radius 10 cm.
A. 10
B. 50
C. 100
D. 200

Answer 1

Hint: We will first draw as given and see that the base will be equal to diameter of circle and height equal to the radius and then put in these values in the formula of area of the triangle and get our answer.

Complete step-by-step answer:
Let us first draw as given in the question:-

               

We see that AC will be a chord of the circle.
We need to maximize the area of the triangle and the area of a triangle is given by $A = \dfrac{1}{2} \times base \times height$.
If the base and height of the triangle are maximum, the area will be maximum as well.
Now, coming back to the saying that AC is a chord. The chord can take maximum length if it is a diameter, because diameter is the longest chord of a circle. Hence, the base is equal to the diameter of the circle.
Since radius is equal to 10 cm. and diameter is twice the radius.
Hence, the diameter of the circle is 20 cm.
So, take base = 20 cm. in the formula required.
Now, if we draw perpendicularly from center to the end of the circle, it will always be equal to radius.
Hence, height = 10 cm.
Putting these values base = 20 cm and height = 10 cm in the formula for area of a triangle that is: $A = \dfrac{1}{2} \times base \times height$. We will get:-
$A = \dfrac{1}{2} \times 10 \times 20 = 10 \times 10 = 100c{m^2}$.
Hence, the maximum area can be 100 square cm.

So, the correct answer is “Option C”.

Note: The students must note that they cannot use the base as diameter without showing that it will lead to the maximum area.
Fun Fact:- If we notice that the area of a triangle is half the area of a rectangle, it is because if we divide a rectangle in two triangles, we will get an area of two triangles.