Question 2 - Figure it out - Page 169-170 - Chapter 7 Class 8 - Area (Ganita Prakash II)

Transcript

Question 2 Give a method to convert a rectangle into a rhombus of equal area using dissection.Let’s look at it Step-by-Step Rectangle to Rhombus A geometric dissection proving area equivalence by physically converting a rectangle into a rhombus. STEP 1 OF 6 The Starting Rectangle Start with a rectangle ABCD . Let the total width be , and the height be h. The area of this rectangle is strictly . Previous Next Step Rectangle to Rhombus A geometric dissection proving area equivalence by physically converting a rectangle into a rhombus. STEP 2 OF 6 Mark and Cut Mark the exact midpoint on the top edge. Draw diagonal lines from M to the bottom left (D) and bottom right (C) corners. This cleanly divides the rectangle into three separate triangles. Previous Next StepRectangle to Rhombus A geometric dissection proving area equivalence by physically converting a rectangle into a rhombus. STEP 3 OF 6 Isolate the Pieces We now have three pieces: A large central isosceles triangle (Indigo). Two identical smaller right-angled triangles on the sides (Blue). Notice the legs of the blue triangles are and w/2. Previous Next StepRectangle to Rhombus A geometric dissection proving area equivalence by physically converting a rectangle into a rhombus. STEP 4 OF 6 The Cross-Slide Now for the magic. We will take the left triangle and slide it down to the bottom right. We will take the right triangle and slide it down to the bottom left. Their vertical edges (height ) will perfectly merge together in the center! Previous Next StepRectangle to Rhombus A geometric dissection proving area equivalence by physically converting a rectangle into a rhombus. STEP 5 OF 6 The Resulting Rhombus The pieces interlock flawlessly to form an inverted isosceles triangle attached to the base of the first one. Because all four outer edges are the hypotenuses of identical right-angled triangles, they are exactly the same length. This shape is a perfect Rhombus! Previous Next Step Rectangle to Rhombus A geometric dissection proving area equivalence by physically converting a rectangle into a rhombus. STEP 6 OF 6 Area Equivalence Proof No pieces were added or removed, so the Area remains exactly the same as the original rectangle. Let's verify this using the Rhombus area formula. Area of Rectangle Rhombus Diagonals: Area of Rhombus