R s aggarwal solutions mathematics class 9 Solutions for Chapter 11 areas of parallelograms and triangles

Exercises in Chapter 11 areas of parallelograms and triangles Grade 9

Areas Of Parallelograms And Triangles Exercise 11

Questions in Areas Of Parallelograms And Triangles Exercise 11

Which of the following figures lie on the same base and between the same parallels. In such a case, write the common base and the two parallels.

Find the area of a figure formed by joining the midpoints of the adjacent sides of a rhombus with diagonals 12 cm and 16 cm.

Find the area of a trapezium whose parallel sides are 9cm and 6cm respectively and the distance between these sides is 8cm.

In the adjoining figure, ABCD is a trapezium in which AB || DC; AB = 7cm; AD = BC = 5cm and the distance between AB and DC is 4cm. Find the length of DC and hence, find the area of the trap. ABCD.

BD is one of the diagonals of a quad. ABCD. If AL ⊥ BD and CM ⊥ BD, show that Ar (quad. ABCD) = ½ × BD × (AL + CM).

M is the midpoint of the side AB of a parallelogram ABCD. If ar (AMCD) = 24 cm^2, find ar (△ ABC).

In the adjoining figure, ABCD is a quadrilateral in which diag. BD = 14cm. If AL ⊥ BD and CM ⊥ BD such that AL = 8cm and CM = 6cm, find the area of quad. ABCD.

If P and Q are any two points lying respectively on the sides DC and AD of a parallelogram ABCD then show that ar (△ APB) = ar (△ BQC).

In the adjoining figure, MNPQ and ABPQ are parallelograms and T is any point on the side BP. Prove that

(i) ar (MNPQ) = ar (ABPQ)

(ii) ar (△ ATQ) = ½ ar (MNPQ).

In the adjoining figure, ABCD is a trapezium in which AB || DC and its diagonals AC and BD intersect at O. Prove that ar (△ AOD) = ar (△ BOC).

In the adjoining figure, DE || BC. Prove that

(i) ar (△ ACD) = ar (△ ABE)

(ii) ar (△ OCE) = ar (△ OBD).

Prove that a median divides a triangle into two triangles of equal area.

Show that a diagonal divides a parallelogram into two triangles of equal area.

In the adjoining figure, ABC and ABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (△ ABC) = ar (△ ABD).

D and E are points on sides AB and AC respectively of △ ABC such that ar (△ BCD) = ar (△ BCE). Prove that DE || BC.

P is any point on the diagonal AC of a parallelogram ABCD. Prove that ar (△ ADP) = ar (△ ABP).

In the adjoining figure, the diagonals AC and BD of a quadrilateral ABCD intersect at O. If BO = OD, prove that ar (△ ABC) = ar (△ ADC).

The vertex A of △ ABC is joined to a point D on the side BC. The midpoint of AD is E. Prove that ar (△BEC) = ½ ar (△ ABC).

D is the midpoint of side BC of △ ABC and E is the midpoint of BD. If O is the midpoint of AE, prove that ar (△ BOE) = 1/8 ar (△ ABC).

In a trapezium ABCD, AB || DC and M is the midpoint of BC. Through M, a line PQ || AD has been drawn which meets AB in P and DC produced in Q, as shown in the adjoining figure. Prove that ar(ABCD) = ar (APQD).

In the adjoining figure, ABCD is a quadrilateral. A line through D, parallel to AC, meets BC produced in P. Prove that ar (△ ABP) = ar (quad. ABCD).

In the adjoining figure, △ ABC and △ DBC are on the same base BC with A and D on opposite sides of BC such that ar (△ ABC) = ar (△ DBC). Show that BC bisects AD.

ABCD is a parallelogram in which BC is produced to P such that CP = BC, as shown in the adjoining figure. AP intersects CD at M. If ar (DMB) = 7 cm2, find the area of parallelogram ABCD.

In a parallelogram ABCD, any point E is taken on the side BC. AE and DC when produced to meet at a point M. Prove that ar (△ ADM) = ar (ABMC).

P, Q, R, S are respectively the midpoints of the sides AB, BC, CD and DA of parallelogram ABCD. Show that PQRS is a parallelogram and also show that ar (||gm PQRS) = ½ × ar (||gm ABCD).

In a triangle ABC, the medians BE and CF intersect at G. Prove that ar (△ BCG) = ar (AFGE).

The base BC of △ ABC is divided at D such that BD = ½ DC. Prove that ar (△ ABD) = 1/3 × ar (△ABC).

In the adjoining figure, BD || CA, E is the midpoint of CA and BD = ½ CA. Prove that ar (△ ABC) = 2 ar (△ DBC).

The given figure shows a pentagon ABCDE. EG, drawn parallel to DA, meets BA produced to G, and CF, drawn parallel to DB, meets AB produced at F. Show that ar (pentagon ABCDE) = ar (△ DGF).

In the adjoining figure, CE || AD and CF || BA. Prove that ar (△ CBG) = ar (△ AFG).

In the adjoining figure, the point D divides the side BC of △ ABC in the ratio m: n. Prove that ar (△ABD): ar (△ ADC) = m: n.

In a trapezium ABCD, AB || DC, AB = a cm, and DC = b cm. If M and N are the midpoints of the nonparallel sides, AD and BC respectively then find the ratio of ar (DCNM) and ar (MNBA).

ABCD is a trapezium in which AB || DC, AB = 16cm and DC = 24cm. If E and F are respectively the midpoints of AD and BC, prove that ar (ABFE) = 9/11 ar (EFCD).

In the adjoining figure, D and E are respectively the midpoints of sides AB and AC of △ ABC. If PQ || BC and CDP and BEQ are straight lines then prove that ar (△ ABQ) = ar (△ ACP).