Question 9 - Figure it out (Page 162-164) - Area of Parallelogram - Chapter 7 Class 8 - Area (Ganita Prakash II)

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Question 9 - Figure it out (Page 162-164) Which has greater area — an equilateral triangle or a square of the same sidelength as the triangle? Which has greater area — two identical equilateral triangles together or a square of the same sidelength as the triangle? Give reasons.Let’s look at it separately One Equilateral Triangle vs. Square Imagine a square with 5 cm sides. Its base is 5 cm, and its vertical height is exactly 5 cm. Now imagine an equilateral triangle with 5 cm sides. Because the sides slant inward to meet at a point, its vertical height is severely reduced (it is strictly less than 5 cm). Since it has the same base but a shorter height, the square has the greater area. Thus, square has more area Let’s look at it visually STEP 1 OF 4 The Square ( 5 cm ) Let's assume a side length of 5 cm . Area of Square Side × Side Area . Previous Next Step STEP 2 OF 4 The Equilateral Triangle Now, let's place an equilateral triangle with the exact same side length ( 5 cm ) right on top of the square's base. Previous Next StepSTEP 3 OF 4 Why is the Height Smaller? If you drop a straight line down from the peak, the 5 cm slanted side becomes the hypotenuse of a right triangle. Since the hypotenuse is ALWAYS the longest side, the straight height MUST be smaller! Math: Height . Previous Next Step STEP 4 OF 4 Conclusion: 1 Triangle vs Square Square Area . Triangle Area Base × Height Triangle Area . . Therefore, the Square has a much greater area. Previous RestartTwo Equilateral Triangles vs. Square Now, A square's area is exactly side × side. The height of an equilateral triangle is only about 86.6% of its side length. Even if you combine the areas of two equilateral triangles, their combined area formula simplifies to approximately 0.866 × (side × side). Because 0.866 is less than 1, the square still has the greater area. When you place the triangle inside the square, it is significantly shorter and leaves plenty of empty space inside the square's bounds. Thus, square has more area Let’s look at it visually STEP 1 OF 5 The Setup (5 cm sides) Let's test two triangles! Inside the Square ( ) sits Triangle 1. On the right sits Triangle 2. Both triangles have 5 cm sides. Previous Next Step STEP 2 OF 5 Chop the Second Triangle To see if the two triangles combined have more area than the square, let's cut Triangle 2 perfectly in half down the middle. Previous Next StepSTEP 3 OF 5 Rearrange the Pieces Now, let's take those two halves, flip them upside down, and slide them into the empty spaces beside Triangle 1. Previous Next Step STEP 4 OF 5 A Perfect Rectangle The two equilateral triangles perfectly merge to form a solid rectangle! The width is 5 cm (the base), and the height is exactly the triangle's height ( 4.33 cm). Previous Next Step STEP 5 OF 5 The Final Calculation Combined Area Base × Height Area . Square Area . . Therefore, 1 Square is greater than 2 Equilateral Triangles! Previous Restart