Perimeter and Area | IT

Question 18

Area Maze Puzzles

In each figure, find the missing value of either the length of a side or the area of a region.

Question diagram 1
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Solution
Understand the Question
  • In an area maze puzzle, every region is a rectangle whose area is given by Area=Length×Breadth\text{Area} = \text{Length} \times \text{Breadth}.
  • When a rectangle is divided into a 2×22 \times 2 grid of smaller rectangles sharing boundaries, the product of the areas of diagonally opposite rectangles is equal: Area1×Area4=Area2×Area3\text{Area}_1 \times \text{Area}_4 = \text{Area}_2 \times \text{Area}_3
  • For composite step-like shapes, find the unknown side lengths sequentially from the known areas and side dimensions.

(a) Find the missing area in figure (a).

Step 1 · Calculate the Missing Area

Diagram 1

Let the missing area be XX.

For a 2×22 \times 2 grid of rectangles inside a larger rectangle, the products of diagonally opposite areas are equal: Area(top-left)×Area(bottom-right)=Area(top-right)×Area(bottom-left)\text{Area}(\text{top-left}) \times \text{Area}(\text{bottom-right}) = \text{Area}(\text{top-right}) \times \text{Area}(\text{bottom-left})

13×X=26×1513X=390X=39013X=30\begin{aligned} 13 \times X &= 26 \times 15 \\[0.6em] 13X &= 390 \\[0.6em] X &= \dfrac{390}{13} \\[0.6em] X &= 30 \end{aligned}
Answer

(a) 30 sq cm30\text{ sq cm}

(b) Find the missing area in figure (b).

Step 1 · Find the Missing Area

For the bottom rectangle: Width=AreaHeight=102=5 cm\text{Width} = \dfrac{\text{Area}}{\text{Height}} = \dfrac{10}{2} = 5\text{ cm}

For the top-right rectangle: Width=3 cm,Height=3 cm\text{Width} = 3\text{ cm}, \quad \text{Height} = 3\text{ cm}

Area=Width×Height=3 cm×3 cm=9 sq cm\begin{aligned} \text{Area} &= \text{Width} \times \text{Height} \\[0.6em] &= 3\text{ cm} \times 3\text{ cm} \\[0.6em] &= 9\text{ sq cm} \end{aligned}
Answer

(b) 9 sq cm9\text{ sq cm}

Common Mistakes
  • Measuring Visually: Area maze puzzles are deliberately not drawn to scale. Never measure with a ruler; always deduce side lengths using Area=Length×Breadth\text{Area} = \text{Length} \times \text{Breadth}.
  • Diagonal Area Property: In a 2×22 \times 2 rectangular grid, remember that diagonally opposite areas multiply to the same product (A1×A4=A2×A3A_1 \times A_4 = A_2 \times A_3), not their sum.

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Q14

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Without calculating all over again, observe, think and find out what will be the change in the perimeter if a new square is attached as shown on the right.

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Q16

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Q18

Area Maze Puzzles

In each figure, find the missing value of either the length of a side or the area of a region.

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