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Question

Which of the following statements are true and which are false;

(i) All squares are congruent.

(ii) If two squares have equal areas, they are congruent.

(iii) If two rectangles have equal area, they are congruent.

(iv) If two triangles are equal in area, they are congruent.

(i) All squares are congruent.

(ii) If two squares have equal areas, they are congruent.

(iii) If two rectangles have equal area, they are congruent.

(iv) If two triangles are equal in area, they are congruent.

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Solution

i) False. All the sides of a square are of equal length. However, different squares can have sides of different lengths. Hence all squares are not congruent.

ii) True

Area of a square = side $\times $side

Therefore, two squares that have the same area will have sides of the same lengths. Hence they will be congruent.

iii) False

Area of a rectangle = length $\times $ breadth

Two rectangles can have the same area. However, the lengths of their sides can vary and hence they are not congruent.

Example: Suppose rectangle 1 has sides 8 m and 8 m and area 64 metre square.

Rectangle 2 has sides 16 m and 4 m and area 64 metre square.

Then rectangle 1 and 2 are not congruent.

iv) False

Area of a triangle = $\frac{1}{2}\times \mathrm{base}\times \mathrm{height}$

Two triangles can have the same area but the lengths of their sides can vary and hence they cannot be congruent.

ii) True

Area of a square = side $\times $side

Therefore, two squares that have the same area will have sides of the same lengths. Hence they will be congruent.

iii) False

Area of a rectangle = length $\times $ breadth

Two rectangles can have the same area. However, the lengths of their sides can vary and hence they are not congruent.

Example: Suppose rectangle 1 has sides 8 m and 8 m and area 64 metre square.

Rectangle 2 has sides 16 m and 4 m and area 64 metre square.

Then rectangle 1 and 2 are not congruent.

iv) False

Area of a triangle = $\frac{1}{2}\times \mathrm{base}\times \mathrm{height}$

Two triangles can have the same area but the lengths of their sides can vary and hence they cannot be congruent.

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