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1.

One diagonal of a rectangle is 18 cm. What is the length of its other diagonal?

Answer»

∵ In a rectangle, diagonals are equal

⇒ AC = BD

Given, one diagonal of a rectangle = 18cm

∴ Other diagonal of a rectangle will be = 18cm

i.e. AC = BD = 18cm.
Discussion
2.

Angles of a quadrilateral are (4x)°, 5(x + 2)°, (7x - 20)° and 6(x + 3)°. Find(i) the value of x.(ii) each angle of the quadrilateral.

Answer»

Angles of quadrilateral are,

(4x)°, 5(x + 2)°, (7x - 20)° and 6(x + 3)°.

4x + 5(x + 2) + (7x - 20) + 6(x + 3) = 360°

4x+5x+10+7x-20+6x+18 = 360° 

22x + 8 = 360°

22x = 360°- 8°

22x = 352°

x = 16°

Hence angles are,

(4x)° = (4 x 16)° = 64°,

5(x + 2)° = 5(16 + 2)° = 90°,

(7x - 20)° = (7 x 16 - 20)° = 92°

6(x + 3)° = 6(16+3) = 114°
Discussion
3.

Two angles of a quadrilateral are 68° and 76°. If the other two angles are in the ratio 5 : 7; find the measure of each of them.

Answer»

Two angles are 68° and 76°

Let other two angles be 5x and 7x

∴ 68°+ 76° + 5x + 7x = 360°

12x + 144° = 360°

12x = 360° – 144°

12x = 216°

x = 18°

angles are 5x and 7x

i.e. 5×18° and 7×18° i.e. 90° and 126°
Discussion
4.

If three angles of a quadrilateral are 90° each, show that the given quadrilateral is a rectangle.

Answer»

The given quadrilateral ABCD will be a rectangle, if its each angle is 90°

Since, the sum of interior angles of a quadrilateral is 360°.

∴∠A +∠B + ∠C + ∠D = 360°

⇒ 90° + 90° + 90° + ∠D = 360°

⇒ 270° + ∠D = 360°

⇒ ∠D = 360° – 270°

⇒ ∠D = 90°

Since, each angle of the quadrilateral is 90°.

∴ The given quadrilateral is a rectangle.
Discussion
5.

The diagonals of a rhombus are 6 .cm and 8 cm. State the angle at which these diagonals intersect.

Answer»

The diagonals of a Rhombus always intersect at 90°.
Discussion
6.

Write, giving reason, the name of the figure drawn alongside. Under what condition will this figure be a square.

Answer»

Since, all the sides of the given figure are equal.

i.e. AB = BC = CD = DA = 6 cm

∴ The given figure is a rhombus.

This figure shall be considered as a square if any angle is 90°.
Discussion
7.

Write two conditions that will make the adjoining figure a square.

Answer»

The conditions that will make the ad-joining figure a square are :

(i) All the sides must be equal.

(ii) Any angle is 90°.
Discussion