Selina solutions concise maths class 9 icse Solutions for Chapter 12 chapter 12 mid point and its converse

Exercises in Chapter 12 chapter 12 mid point and its converse Grade 9

Exercise 12(A)

Exercise 12(B)

Questions in Exercise 12(A)

In triangle ABC, M is mid-point of AB and a straight line through M and parallel to BC

cuts AC at N. Find the lengths of AN and MN if BC = 7cm and AC = 5cm.

Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle

is a rhombus.

D, E, and F are the mid-points of the sides AB, BC, and CA of an isosceles triangle ABC in

which AB=BC. Prove that triangle DEF is also isosceles.

The following figure shows a trapezium ABCD in which AB||DC. P is the mid-point of AD and

PR||AB. Prove that:

The figure given below shows a trapezium ABCD. M and N are the mid-points of the nonparallel

sides AD and BC respectively. Find:

(i) MN, if AB = 11 cm and DC = 8 cm

(ii) AB, if DC = 20 cm and MN = 27 cm

(iii) DC, if MN = 15 cm and AB = 23 cm

The diagonals of a quadrilateral intersect at right angles. Prove that the figure obtained by

joining the mid-points of the adjacent sides of the quadrilateral is a rectangle.

L and M are the mid-points of sides AB and DC respectively of parallelogram ABCD.

Prove that segments DL and BM trisect diagonal AC.

ABCD is a quadrilateral in which AD = BC. E, F, G, and H are the mid-points of AB, BD,

CD and AC respectively. Prove that EFGH is a rhombus.

D, E, and F are the mid-points of the sides AB, BC, and CA respectively of triangle ABC. AE

meets DF at O. P and Q are the mid-points of OB and OC respectively. Prove that DPQF is

a parallelogram.

In a triangle ABC, P is the mid-point of side BC. A line through P and parallel to CA meets

AB at point Q; and a line through Q and parallel to BC meet median AP at point R. Prove

that:

(i) AP = 2AR

(ii) BC = 4QR

In trapezium ABCD, AB is parallel to DC; P and Q are the mid-points of AD and BC

respectively. BP produced meets CD produced at point E. Prove that:

(i) Point P bisects BE,

(ii) PQ is parallel to AB.

In a triangle ABC, AD is a median and E is the mid-point of median AD. A line through B and

E meets AC at point F.

Prove that: AC = 3AF

D and F are mid-points of sides AB and AC of a triangle ABC. A line through F and

parallel to AB meets BC at point E.

(i) Prove that BDFE is a parallelogram.

(ii) Find AB, if EF = 4.8 cm.

In triangle ABC, AD is the median and DE, drawn parallel to side BA, meets AC at point

E. Show that BE is also a median.

In triangle ABC, AD is the median and DE, drawn parallel to side BA, meets AC at point

E. Show that BE is also a median.

In the given figure, M is the mid-point of AB and DE, whereas N is the mid-point of BC and DF.

Show that: EF = AC.

Questions in Exercise 12(B)

Use the following figure to find:

(i) BC, if AB = 7.2 cm.

(ii) GE, if FE = 4 cm.

(iii) AE, if BD = 4.1 cm.

(iv) DF, if CG = 11 cm.

In the figure, given below, 2AD = AB, P is mid-point of AB, Q is mid-point of DR and

PR||BS. Prove that:

(i) AQ || BS,

(ii) DS = 3RS

The side AC of a triangle ABC is produced to point E so that $\mathrm{CE}=\frac{1}{2} \mathrm{AC}$ D is the mid-point

of BC and ED produced meets AB at F. Lines through D and C are drawn parallel to AB

which meet AC at point P and EF at point R respectively. Prove that:

(i) 3DF = EF

(ii) 4CR = AB

In triangle ABC, the medians BP and CQ are produced up to points M and N respectively

such that BP = PM and CQ = QN. Prove that:

(i) M, A, and N are collinear.

(ii) A is the mid-point of MN.

In triangle ABC, angle B is obtuse. D and E are mid-points of sides AB and BC respectively

and F is a point on side AC such that EF is parallel to AB. Show that BEFD is a

parallelogram.

In parallelogram ABCD, E and F are mid-points of the sides AB and CD respectively. The

line segments AF and BF meet the line segments ED and EC at points G and H

respectively.

Prove that:

(i) Triangles HEB and FHC are congruent;

(ii) GEHF is a parallelogram.

In triangle ABC, D and E are points on side AB such that AD = DE = EB. Through D and

E, lines are drawn parallel to BC which meet side AC at points F and G respectively.

Through F and G, lines are drawn parallel to AB which meets side BC at points M and N

respectively. Prove that: BM = MN = NC.

In triangle ABC; M is the mid-point of AB, N is the mid-point of AC and D is any point in the base

BC. Use Intercept Theorem to show that MN bisects AD.

If the quadrilateral formed by joining the mid-points of the adjacent sides of a quadrilateral

ABCD is a rectangle, show that the diagonals AC and BD intersect at a right angle.

In triangle ABC; D and E are mid-points of the sides AB and AC respectively. Through E,

a straight line is drawn parallel to AB to meet BC at F. Prove that BDEF is a parallelogram.

If AB = 16 cm, AC = 12 cm and BC = 18 cm, find the perimeter of the parallelogram

BDEF.

In parallelogram ABCD, E is the mid-point of AB and AP is parallel to EC which meets

DC at point O and BC produced at P. Prove that:

(i) BP = 2AD

(ii) O is the mid-point of AP.

In trapezium ABCD, sides AB and DC are parallel to each other. E is the mid-point of AD and

F is the mid-point of BC.

Prove that: AB + DC = 2EF

In triangle ABC, AD is the median and DE is parallel to BA, where E is a point in AC.

Prove that BE is also a median.

Adjacent sides of a parallelogram are equal and one of the diagonals is equal to any one of

the sides of this parallelogram. Show that its diagonals are in the ratio√𝟑: 𝟏.

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Chapters in this book

Chapter 1- Rational and Irrational Numbers

Chapter 2- Compound Interest [Without Using Formula

Chapter 3- Compound Interest [Using Formula

Chapter 4- Expansions

Chapter 5- Factorisation

Chapter 6- Simultaneous Equations

Chapter 7- Indices [exponents]

Chapter 8- Logarithms

Chapter 9- Triangles [Congruency in Triangles]

Chapter 10- Isosceles Triangle

Chapter 11- Inequalities

Chapter 12- Mid-Point and Its Converse

Chapter 13- Pythagoras Theorem

Chapter 14- Rectilinear Figures

Chapter 15- Construction of Polygons

Chapter 16- Area Theorems

Chapter 17- Circles

Chapter 18- Statistics

Chapter 19- Mean and Median

Chapter 20- Area and Perimeter of Plane Figures

Chapter 21- Solids

Chapter 22- Trigonometrical Ratios

Chapter 23- Trigonometrical Ratios of Standard Angles

Chapter 24- Solution Of Right Triangles

Chapter 25- Complementary angles

Chapter 26- Co-ordinate Geometry

Chapter 27- Graphical Solution

Chapter 28- Distance Formula