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Geometry

Quadrilaterals

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CAT 2025 Lesson : Quadrilaterals - Symmetrical Shapes

4. Areas with Symmetrical Shapes

In these questions we will be required to find the area of a shaded region when the overall area is given. Or, we might be required to find the ratio of the area of shaded region to that of the unshaded region, etc. We look for patterns in these shapes or divide them into symmetrical parts.

Example 22

In the following diagram, ABCD is a rectangle where E and F are the mid-points of AD and BC respectively. What percentage of the rectangle is shaded?

Solution

We mark G and F – the mid-points of AB and CD respectively.

When we join BD and GH, we get 8 congruent triangles.

Of these 8 triangles, only 2 of them are shaded.

∴ Portion of shaded area

= \dfrac{2}{8} \times 100 \% = 25 \%

Answer:

25 \%

Example 23

In the below figure, H, E, D, F and G are the mid-points of BC, AC, AB, BD and EC respectively. What is the ratio of the shaded region to the unshaded region?

Solution

The parallelograms DEOF and FOBH are congruent.

∴ If we move the shaded region from DEOF to FOBH, we get the figure on the right.

\angle

A is common and

\mathrm {\dfrac{AF}{AB} = \dfrac{AG}{AC}= \dfrac{3}{4}}

∴ By SAS similarity rule,

\triangle

AFG

\sim

\triangle

ABC

\mathrm{ \dfrac{\triangle AFG}{\triangle ABC} = \dfrac{AF^2}{AB^2} = \dfrac{9}{16}}

Area of shaded region to that of

\triangle

ABC = 1

- \dfrac{9}{16} = \dfrac {7}{16}

Ratio of areas of shaded to unshaded = 7 : 9

Answer:

7 : 9

Example 24

ABCD is a rectangle. P and Q are points on CD such that DP = PQ = QC. O is a point on AB. What is the ratio of the shaded region to that of ABCD?

Solution

Area of rectangle ABCD

= \mathrm {length \times breadth} = 3xy

\triangle

OPQ, the altitude to base PQ will be of the same length as AD, i.e.,

y

.

Area of

\triangle

OPQ

= \dfrac{1}{2} \times y \times x = \dfrac{xy}{2}

Ratio of

\triangle

OPQ to ABCD

= \dfrac{xy}{2} : 3 xy = 1 : 6

Answer:

1 : 6

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