M l aggarwal understanding icse mathematics class 9 Solutions for Chapter 13 rectilinear figures

Exercises in Chapter 13 rectilinear figures Grade 9

Rectilinear Figures Exercise 13.1

Rectilinear Figures Exercise 13.2

Questions in Rectilinear Figures Exercise 13.1

If two angles of a quadrilateral are 40° and 110° and the other two are in the ratio 3 : 4, find

these angles.

If the angles of a quadrilateral, taken in order, are in the ratio 1: 2 : 3: 4, prove that it is a

trapezium.

(a) In figure (1) given below, ABCD is a parallelogram in which ∠DAB = 70°, ∠DBC = 80°.

Calculate angles CDB and ADB.

(b) In figure (2) given below, ABCD is a parallelogram. Find the angles of the AAOD.

Given, ABCD is a rectangle

We know that the diagonals of a rectangle are the same

and bisect each other

So, we have

AP = BP

(i) Prove that each angle of a rectangle is 90°.

(ii) If the angle of a quadrilateral is equal, prove that it is a rectangle.

(iii) If the diagonals of a rhombus are equal, prove that it is a square.

(iv) Prove that every diagonal of a rhombus bisects the angles at the vertices.

(i) Prove that bisectors of any two adjacent angles of a parallelogram are at right angles.

(ii) Prove that bisectors of any two opposite angles of a parallelogram are parallel.

(iii) If the diagonals of a quadrilateral are equal and bisect each other at right angles, then prove

that it is a square.

(a) In figure (1) given below, ABCD is a parallelogram and X is the mid-point of BC. The line AX

produced meets DC produced at Q. The parallelogram ABPQ is completed.

Prove that:

(i) the triangles ABX and QCX are congruent;

(ii)DC = CQ = QP

(b) In figure (2) given below, points P and Q have been taken on opposite sides AB and CD

respectively of a parallelogram ABCD such that AP = CQ. Show that AC and PQ bisect each

other.

If P and Q are points of trisection of the diagonal BD of a parallelogram ABCD, prove that CQ

|| AP.

.A transversal cuts two parallel lines at A and B. The two interior angles at A are bisected and

so are the two interior angles at B; the four bisectors form a quadrilateral ABCD. Prove that

(i) ABCD is a rectangle.

(ii) CD is parallel to the original parallel lines.

In a parallelogram ABCD, the bisector of ∠A meets DC in E and AB = 2 AD. Prove that

(i) BE bisects ∠B

(ii) ∠AEB = a right angle.

In a parallelogram ABCD, the bisector of ∠A meets DC in E and AB = 2 AD. Prove that:

(i) BE bisects ∠B

(ii) ∠AEB is a right angle

ABCD is a square and the diagonals intersect at O. If P is a point on AB such that AO =AP,

prove that 3 ∠POB = ∠AOP.

ABCD is a square. E, F, G and H are points on the sides AB, BC, CD and DA respectively such

that AE = BF = CG = DH. Prove that EFGH is a square.

ABCD is a rhombus in which ∠A = 60°. Find the ratio AC : BD.

Questions in Rectilinear Figures Exercise 13.2

Using ruler and compasses only, construct the quadrilateral ABCD in which ∠ BAD = 45°, AD =

AB = 6cm, BC = 3.6cm, CD = 5cm. Measure ∠ BCD.

Using ruler and compasses only, construct the quadrilateral ABCD given that AB = 5 cm, BC =

2.5 cm, CD = 6 cm, ∠BAD = 90° and the diagonal AC = 5.5 cm.

Construct a trapezium ABCD in which AD || BC, ∠B= 60°, AB = 5 cm. BC = 6.2 cm and CD =

4.8 cm.

Using ruler and compasses only, construct a parallelogram ABCD with AB = 5.1 cm, BC = 7 cm

and ∠ABC = 75°.

Using ruler and compasses, construct a parallelogram ABCD give that AB = 4 cm, AC = 10 cm,

BD = 6 cm. Measure BC.

Using ruler and compasses only, construct a parallelogram ABCD such that BC = 4 cm,

diagonal AC = 8.6 cm and diagonal BD = 4.4 cm. Measure the side AB.

Using ruler and compasses only, construct a parallelogram ABCD with AB = 6 cm, altitude =

3.5 cm and side BC = 4 cm. Measure the acute angles of the parallelogram.

The perpendicular distances between the pairs of opposite sides of a parallelogram ABCD are

3 cm and 4 cm and one of its angles measures 60°. Using a ruler and compasses only, construct

ABCD.

Using ruler and compasses, construct a rectangle ABCD with AB = 5cm and AD = 3 cm.

Using ruler and compasses only construct A rhombus ABCD given that AB 5cm, AC = 6cm

measure ∠BAD.

Construct a rhombus PQRS whose diagonals PR and QS are 8cip and 6cm respectively.

Construct a rhombus ABCD of side 4.6 cm and ∠BCD = 135°, by using a ruler and compasses

only.

Construct a trapezium ABCD when one of parallel sides AB = 4.8 cm, height = 2.6cm, BC = 3.1

cm and AD = 3.6 cm.

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Chapters in this book

Rational and Irrational Numbers

Compound Interest

Expansions

Factorization

Simultaneous Linear Equations

Problems on Simultaneous Linear Equations

Quadratic Equations

Indices

Logarithms

Triangles

Mid Point Theorem

Pythagoras Theorem

Rectilinear Figures

Theorems on Area

Circle

Mensuration

Trigonometric Ratios

Trigonometric Ratios and Standard Angles

Coordinate Geometry

Statistics