Question 26 - End-of-Chapter Exercises - Chapter 6 Class 9 - Measuring Space: Perimeter and Area (Ganita Manjar

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Question 26 In Fig. 6.54, we see three triangles within a rectangle. The areas of the triangles are 𝐴,𝐵,𝐶, as marked. Show that the area of the rectangle is (2(𝐴+𝐶)(𝐵+𝐶))/𝐶. Let’s answer this step by step Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C 1 Here is the original figure. We have a large yellow rectangle containing three colored triangles: A (blue), B (green), and C (pink Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 2 Let the large rectangle have a total width W and total height H. The area we eventually want to find is W×H Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C 3 Look at Triangle C (pink). It's a right-angled triangle. Let's label its vertical side h (red) and its horizontal side w (blue)Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 4 The area of a right triangle is 1/2× base × height. So, the area of Triangle C is: C=1/2×w×h Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 5 Now, let's group Triangles A and C together. Notice how they both share that same vertical red edge h ? Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 6 If you treat that shared edge h as their 'base', the total horizontal distance they stretch across is exactly the width of the rectangle, W Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 7 Therefore, their combined area ( 1/2× base × stretch) is: (A+C)=1/2×W×h. Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 8 Next, let's group Triangles B and C together. They share the horizontal blue edge w Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 9 Treating w as their base, their combined vertical stretch goes all the way from top to bottom, exactly equal to the height H Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 10 So, their combined area is: (B+C)=1/2×H×w Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 8 Next, let's group Triangles B and C together. They share the horizontal blue edge w Problem 26: Triangles in a Rectangle Show that the area of the rectangle is 2(A+C)(B+C)/C. 13 The '1/2', the ' w ', and the ' h ' all cancel out from the top and bottom! We are left with exactly W×H, which is the Area of the Rectangle. Proved! Proof Complete! Great job!