Question 23 - End-of-Chapter Exercises - Chapter 6 Class 9 - Measuring Space: Perimeter and Area (Ganita Manjar

Transcript

Question 23 In Fig. 6.51 we see two concentric circles with a common centre O. A chord BC of the larger circle is drawn, touching the smaller circle at A. The length of BC is 𝑙. Show that the area of the green region enclosed between the two circles is 1/4 𝜋𝑙^2. Let’s answer this step by step Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 1 Here we have two concentric circles (circles with the common center O ). The green ring is the area we want to find. Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 2 Let the radius of the large circle be and the small circle be . The area of the green ring is Area Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 3 A chord BC of length is drawn on the large circle, touching the small circle at point A. This means is a tangent to the small circle. Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 4 Draw a line from center O to point A. A radius drawn to a tangent is always perpendicular . So, and .Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 5 Draw a line from to . This is the radius of the large circle, so . We now have a right-angled triangle OAB . Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 6 A perpendicular line from the center to a chord bisects (cuts in half) the chord. This means is exactly half of . So, Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 7 Let's use the Pythagorean theorem on triangle OAB: Base ^2+ Height ^2= Hypotenuse ^2. This gives us: (I/2)^2+r^2=R^2 Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 8 Let's rearrange this equation by subtracting r^2 from both sides: R^2-r^2=(l/2)^2=l^2/4. Problem 23: Concentric Circles Find the area of the green region enclosed between two concentric circles. 9 Remember our area formula from Step 2? Area =Π(R^2-r^2 ). Substituting what we just found, we get: Area =n(l^2/4)=1/4πl^2. Proved! Proof Complete! Great job!