Question 6 - Figure it out - Page 157-159 - Chapter 7 Class 8 - Area (Ganita Prakash II)

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Question 6 ABCD, BCEF, and BFGH are identical squares. (i) If the area of the red region is 49 sq. units, then what is the area of the blue region?Let’s assume side of each square is s Area of Red Shape Red Shape is a right angled triangle ∆ DCH with Base = CD = s Height = HC = s + s = 2s Area = 49 sq units Putting in formula Area of ∆ DCH = 1/2 × Base × Height 49 = 𝟏/𝟐 × s × 2s 49 = s2 s2 = 49 s2 = 72 s = 7 units Area of Blue Shape Blue Shape is a right angled triangle ∆ DCH with Height = AD = s Now, for base AP Look at the slanted diagonal line. This line goes from the bottom-left corner to the very top right, traveling up a height of 2 squares and across a width of 1 square. Because it moves at a constant slant, by the time it reaches halfway up (the top edge of the first square), it must have traveled exactly halfway across. So, the base of the blue triangle is exactly half a square's width (length = 𝒔/𝟐) Thus, Base = AP = 𝒔/𝟐 Now, Area of Blue Region = Area of ∆ APD = 1/2 × Base × Height = 𝟏/𝟐 × 𝒔/𝟐 × s = 𝒔^𝟐/𝟒 Putting s = 7 = 7^2/4 = 𝟒𝟗/𝟒 square units Question 6 ABCD, BCEF, and BFGH are identical squares. (ii) In another version of this figure, if the total area enclosed by the blue and red regions is 180 sq. units, then what is the area of each square? Red Area = 𝒔^𝟐 Blue Area = 𝒔^𝟐/𝟒 Given Total area is 180 sq units Putting values Total Area = Red Area + Blue Area 180 = 𝒔^𝟐+𝒔^𝟐/𝟒 180 = 𝑠^2 × (1+1/4) 180 = 𝑠^2 × ((4 × 1 + 1)/4) 180 = 𝑠^2 × ((4 + 1)/4) 180 = 𝒔^𝟐 ×𝟓/𝟒 180 × 4/5 = 𝑠^2 36 × 4 = 𝒔^𝟐 144 = 𝑠^2 𝒔^𝟐 = 144 𝑠^2 = 122 s = 12 units