Question 5 - Figure it out - Page 169-170 - Chapter 7 Class 8 - Area (Ganita Prakash II)

Transcript

Question 5 Here is one of the ways to convert trapezium ABCD into a rectangle EFGH of equal area — Given the trapezium ABCD, how do we find the vertices of the rectangle EFGH? [Hint: If ΔAHI ≅ ΔDGI and ΔBEJ ≅ ΔCFJ, then the trapezium and rectangle have equal areas.]Let’s first construct it, and to prove Areas equal – we follow the hint Trapezium to Rectangle A geometric dissection converting a trapezium into a rectangle to prove the area formula. STEP 1 OF 10 The Original Trapezium Start with trapezium ABCD . Let the top base be a and the bottom base be . The height is . Our goal is to construct a rectangle of exactly the same area. Previous Next Step Trapezium to Rectangle A geometric dissection converting a trapezium into a rectangle to prove the area formula. STEP 2 OF 10 Finding the Midpoints Find the exact midpoint I on the left edge AD , and midpoint J on the right edge BC . Draw straight, vertical lines down through these midpoints. Previous Next Step Trapezium to Rectangle A geometric dissection converting a trapezium into a rectangle to prove the area formula. STEP 3 OF 10 The Target Rectangle These vertical lines form the boundaries of our target rectangle EFGH. Notice the pieces of the trapezium sticking out at the bottom ( and ), and the empty spaces missing from the rectangle at the top ( and ). Previous Next Step Trapezium to Rectangle A geometric dissection converting a trapezium into a rectangle to prove the area formula. STEP 4 OF 10 Proving Congruence Why will they fit perfectly? Focus on the left edge. Because is the midpoint, the segment AI = ID (indicated by the tick marks). The top and bottom bases are parallel, meaning their alternating inner angles are equal. By geometric rules (ASA), this proves that the bottom piece is exactly identical in size and shape to the empty top space ΔAHI! Previous Next Step Trapezium to Rectangle A geometric dissection converting a trapezium into a rectangle to prove the area formula. STEP 5 OF 10 Conservation of Area Since the triangles are identical, . Total Trapezium Area Purple Center Bottom Triangles. Target Rectangle Area (Purple Center) Top Empty Spaces. Because the pieces are equal, replacing the bottom pieces with the top spaces guarantees the total area will not change! Previous Next Step