Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
83 views
in Perimeter and Area of Plane Figures by (48.0k points)
closed by

Find the ratio of the area of a square inscribed in a semi-circle of radius r to the area of another square inscribed in the entire circle of radius r.

(a) 2 : 1

(b) 3 : 2

(c) 2 : 5

(d) 3 : 5

1 Answer

+1 vote
by (49.1k points)
selected by
 
Best answer

Correct option is (c) 2 : 5

Let ABCD be the square inscribed in the semi-circle with center O, and side CD the diameter of the semi-circle.

Let the point M be the mid-point of AB.

Now, join OB and OM to get the right △OMB with OB as the hypotenuse.

Therefore, OM2 + MB2 = OB2

OM = side of the square inscribed in the semi-circle,

MB = half of the side; and

OB = radius

Let the side of the square be x.

\(x^2 + (\frac x2)^2 = r^2\)

\(x^2 = \frac{4r^2}5\)

Hence, the area of the semi-circle = \(\frac{4r^2}5\)

Now diagonal of the square inscribed in the circle =2r

Therefore, its area = \(\frac{(2r)^2}{2} = 2r^2\)

Area of the square = \(\frac{\text{diagonal}^2}2\)

Hence, the required ratio = \(\frac{4r^2}5 : 2r^2\) 

\(= \frac 25 : 1\)

\(= 2:5\)

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...