- Diagonals of a rectangle ABCD intersect at point O. If AC = 8 cm then find BO and if…
- In a rhombus PQRS if PQ = 7.5 then find QR. If ∠QPS = 75° then find the measure of ∠PQR…
- Diagonals of a square IJKL intersects at point M, Find the measures of ∠IMJ, ∠JIK and…
- Diagonals of a rhombus are 20 cm and 21 cm respectively, then find the side of rhombus…
- 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.
