If an isosceles triangle ABC, in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm, find the area of the triangle
Solution:
Given, ABC is an isosceles triangle
The measure of sides AB = AC = 6 cm
ABC is inscribed in a circle of radius 9 cm.
We have to find the area of the triangle.
From the figure,
O is the centre of the circle
Join OB and OC.
Let M be the midpoint of BC.
So, OM ⟂ BC
In an isosceles triangle, the median from the vertex is perpendicular to the base.
Since, ABC is an isosceles triangle and M is the midpoint of BC.
AM ⟂ BC
Let AM = x
Let MB = CM = y
Considering triangle AMB,
AMB is a right triangle with M at right angle.
AB² = AM² + BM²
(6)² = x² + y²
x² + y² = 36 ---------------------------- (1)
Considering triangle OMB,
OMB is a right triangle with M at right angle.
OB² = OM² + BM²
OB = OC = 9 (radius of the circle)
From the figure,
AM + MO = AO
AO = radius of circle
x + OM = 9
OM = 9 - x
So, (9)² = (9 - x)² + y²
(a - b)² = a² - 2ab + b²
Now, 81 = 81 - 18x + x² + y²
x² + y² = 18x ----------------------------- (2)
Comparing (1) and (2),
36 = 18x
x = 36/18
x = 2 cm
So, AM = 2 cm
Put x = 2 in (1),
(2)² + y² = 36
4 + y² = 36
y² = 36 - 4
y² = 32
Taking square root,
y = 4√2 cm
Area of triangle = (1/2) × base × height
Area of triangle ABC = (1/2) × BC × AM
BC = BM + CM
= 4√2 + 4√2
BC = 8√2 cm
Area of triangle ABC = (1/2) × 8√2 × 2
= 8√2 square cm
Therefore, the area of the triangle is 8√2 square cm.
✦ Try This: Prove that the tangent to the circumcircle of an isosceles ΔABC at A, in which AB = AC, is parallel to BC.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 10
NCERT Exemplar Class 10 Maths Exercise 9.4 Problem 13
If an isosceles triangle ABC, in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm, find the area of the triangle
Summary:
If an isosceles triangle ABC, in which AB = AC = 6 cm, is inscribed in a circle of radius 9 cm, the area of the triangle is 8√2 square cm
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