Area of the largest triangle that can be inscribed in a semi-circle of radius r units is
a. r² square units
b. 2r square units
c. 1/2 r² square units
d. r²/4 square units
Solution:
Given, radius of semicircle is r units.
We have to find the area of the largest triangle that can be inscribed in semicircle.
Consider a semicircle.
AB is the diameter
AB = 2r units
Let the largest circle inscribed inside the semicircle is ABC.
So, the sides of the triangle AB = 2r
CD = r
We know that the angle in a semi circle is always equal to 90°
So, ∠ACB = 90°
Considering triangle ABC,
ACB is a right triangle with C at right angle.
Area of triangle = (1/2) × base × height
Area of triangle ACB = (1/2) × AB × CD
= (1/2) × 2r × r
= r × r
= r² square units
Therefore, the area of a largest triangle that can be inscribed in a semicircle of radius r is r² square units.
✦ Try This: Find the area of the largest triangle that can be inscribed in a semicircle of radius 5 cm.
Given, radius of semicircle = 5 cm
We have to find the area of the largest triangle that can be inscribed in a semicircle.
Consider a semicircle
AC is the diameter
AC = 2(radius) = 2(5) = 10 cm
Let the triangle inscribed inside the semicircle be ABC
By the properties of circles,
∠ABC = 90°
Area of triangle = (1/2) × base × height
Area of triangle ABC = (1/2) × AC × CO
= (1/2) × 10 × 5
= 5(5)
= 25 square cm.
Therefore, the largest area of the triangle that can be inscribed in a semicircle of radius 5 cm is 25 square cm.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 12
NCERT Exemplar Class 10 Maths Exercise 11.1 Problem 4
Area of the largest triangle that can be inscribed in a semi-circle of radius r units is a. r² square units, b. 2r square units, c. 1/2 r² square units, d. r²/4 square units
Summary:
Area of the largest triangle that can be inscribed in a semi-circle of radius r units is r² square units
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