Question 25 - End-of-Chapter Exercises - Chapter 6 Class 9 - Measuring Space: Perimeter and Area (Ganita Manjar

Transcript

Question 25 Fig. 6.53 shows two circles passing through each other's centres. Find the area of the region enclosed by the two circles in terms of the common radius r. Let’s answer this step by step Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 1 We have two identical circles. Let's label their centers and B. Because they pass through each other's centers, the distance AB is exactly the radius, . Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 2 Let's label the top intersection C and the bottom intersection D. Draw lines connecting A and B to C.Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 3 Look at triangle . The lines and are radii of the circles, so they are both length . AB is also . Since all sides are equal, it's an equilateral triangle! The angle at A is . Problem 25: Intersecting Circles Find the area of the region enclased by two circles passing through each other's centres. 4 The bottom triangle ABD is also equilateral ( ). This means the total angle of the 'pizza slice' (Sector CAD) from center A is .Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 5 The area of Sector CAD is (which is ) of a full circle. Sector Area . Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 6 The dark red shape we want is made by overlapping two of these sectors: Sector CAD (from circle A) and Sector CBD (from circle B).Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 7 If we just add the two sectors together, we have a problem: we count the diamond shape in the middle (ACBD) TWICE. Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 8 To get the correct area, we add the two sectors and subtract the extra diamond. Total Area Sector 1 (Sector 2) - (Diamond ACBD).Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 9 The diamond ACBD is made of 2 equilateral triangles. The area of one equilateral triangle is (√ 3/4)r^2, so two of them equals (√3/2)r^2. Problem 25: Intersecting Circles Find the area of the region enclosed by two circles passing through each other's centres. 10 Now plug it all in: Total Area =(1/3nr^2 )+(1/3├ nr^2 )-(√ 3/2)r^2 Total Area =r^2 (2n/3-√ 3/2) Proof Complete! Great job!