Question 27 - End-of-Chapter Exercises - Chapter 6 Class 9 - Measuring Space: Perimeter and Area (Ganita Manjar

Transcript

Question 27 In the figure we see two shaded regions formed by a quarter circle, a semicircle, and a triangle. Show that the areas of the two shaded regions are equal. Let’s answer this step by step Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 1 Here is the original figure. Notice the two shaded regions: Triangle AOB, and the crescent shape (called a 'Lune') bordered by arcs E and F. Our goal is to prove their areas are exactly equal. Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 2 Let the radius of the main semicircle be r. This means OA=OB=OC=r. The area of the shaded right-angled Triangle AOB is: 1/2 × base × height =1/2r^2 Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 3 Now look at the Quarter Circle AOB (the pie slice from A to O to B). Its area is one-fourth of a full circle: 1/4nr^2.Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 4 Next, let's look at Semicircle E. Its straight diameter is the hypotenuse AB. Using Pythagoras ( r^2+r^2=AB^2 ), we get AB=rV. So its radius is half of that: (r/2)/2 Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 5 Let's calculate the area of Semicircle E: ■(&1/2×n×((r√ 2)/2)^2@& =1/2×n×(2r^2/4)@& =1/4nr^2 ) Wow! Notice it's the EXACT SAME area as the Quarter Circle! Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 6 Look at the unshaded 'White Segment' between the straight line AB and Arc F . How do we find its area? White Segment = (Quarter Circle) - (Triangle AOB). Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 7 Now look at our shaded Lune shape. How is it formed? Lune Area (Semicircle E) (White Segment). Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 8 Let's combine our formulas by substituting the White Segment: Lune Area Semicircle E - [ (Quarter Circle) - (Triangle AOB) ].Problem 27: The Lune and the Triangle Show that the areas of the two shaded regions are equal. 9 Because Semicircle E and the Quarter Circle have the EXACT SAME area (from Step 4), they completely cancel each other out! We are left with just: Lune Area Triangle AOB Area. Proved! Proof Complete! Great job!