Practice Set 5.3, Chapter 5 - Quadrilaterals Mathematics Part II Solutions for Class 9 Math

Practice Set 5.3

1. Diagonals of a rectangle ABCD intersect at point O. If AC = 8 cm then find BO and if…
2. In a rhombus PQRS if PQ = 7.5 then find QR. If ∠QPS = 75° then find the measure of ∠PQR…
3. Diagonals of a square IJKL intersects at point M, Find the measures of ∠IMJ, ∠JIK and…
4. Diagonals of a rhombus are 20 cm and 21 cm respectively, then find the side of rhombus…
5. State with reasons whether the following statements are ‘true’ or ‘false’. (i) Every…

Practice Set 5.3

Question 1.

Diagonals of a rectangle ABCD intersect at point O. If AC = 8 cm then find BO and if ∠CAD = 35° then find ∠ACB.

Answer:

The diagonals of a rectangle are congruent to each other and bisects each other at the point of intersection so since AC = 8 cm

⇒ BD = 8 cm and

O is point of intersection so DO = OB = AO = OC = 4 cm

∠CAD = 35 ° given

⇒ ∠ACB = 35 °

(since AB ∥ DC and AC is transversal ∴ ∠CAD and ∠ACB are pair of alternate interior angle.)

Question 2.

In a rhombus PQRS if PQ = 7.5 then find QR. If ∠QPS = 75° then find the measure of ∠PQR and ∠SRQ.

Answer:

Given quadrilateral is a rhombus.

⇒ all the sides are congruent /equal

⇒ PQ = QR = 7.5

Also ∠QPS = 75° (given)

⇒∠QPS = 75° (opposite angles are congruent)

But ∠QPS + ∠PQR = 180° (adjacent angles are supplementary)

⇒ ∠PQR = 105°

∴ ∠SRQ = 105° (opposite angles)

Question 3.

Diagonals of a square IJKL intersects at point M, Find the measures of ∠IMJ, ∠JIK and ∠LJK.

Answer:

The given quadrilateral is a square

⇒ all the angles are 90°

∴ ∠JIK = 90°

Since the diagonals are  to each other ∠IMJ = 90°

Since the diagonals os a square are bisectors of the angles also

∠LJK = ∠IJL = 1/2 × 90° = 45°

Question 4.

Diagonals of a rhombus are 20 cm and 21 cm respectively, then find the side of rhombus and its perimeter.

Answer:

Let the diagonal AC = 20cm and BD = 21

AB² = BO² + AO²

AB² = (10.5)² + (10)²

(the diagonals of a rhombus bisect each other at 90°)

AB² = 110.25 + 100

AB = √210.25 = 14.5cm (side of the rhombus)

Perimeter = 4a = 14. 5 × 4 = 58cm

Question 5.

State with reasons whether the following statements are ‘true’ or ‘false’.

(i) Every parallelogram is a rhombus.

(ii) Every rhombus is a rectangle.

(iii) Every rectangle is a parallelogram.

(iv) Every square is a rectangle.

(v) Every square is a rhombus.

(vi)Every parallelogram is a rectangle.

Answer:

(i) False.

Explanation: Every Parallelogram cannot be the rhombus as the diagonals of a rhombus bisects each other at 90° but this is not the same with every parallelogram. Hence the statement if false.

(ii) False.

Explanation: In a rhombus all the sides are congruent but in a rectangle opposite sides are equal and parallel. Hence the given statement is false.

(iii) True.

Explanation: The statement is true as in a rectangle opposite angles and adjacent angles all are 90°. And for any quadrilateral to be parallelogram the opposites angles should be congruent.

(iv) True.

Explanation: Every square is a rectangle as all the angles of the square at 90° , diagonal bisects each other and are congruent , pair of opposite sides are equal and parallel . Hence every square is a rectangle is true statement.

(v) True.

Explanation: The statement is true as all the test of properties of a rhombus are meet by square that is diagonals are perpendicular bisects each other , opposite sides are parallel to each other and the diagonals bisects the angles.

(vi) False.

Explanation:

Every parallelogram is a rectangle is not true as rectangle has each angle of 90° measure but same is not the case with every parallelogram.