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

If Fig. 15.26, ABCD is a parallelogram, AE ⊥ DC and CF ⊥ AD. If AB=16 cm, AE=8 cm and CF=10 cm, find AD.

In Q. No. 1, if AD=6 cm, CF=10 cm and AE=8 cm, find AB.

Let ABCD be a parallelogram of area 124 cm². If E and F are the mid-points of sides AB and CD respectively, then find the area of parallelogram AEFD.

If ABCD is a parallelogram, then prove that

ar(Δ ABD) = ar(Δ BCD) = ar(Δ ABC)=ar(Δ ACD) = ar(||^gm ABCD)

In Fig. 15.74, compute the aresa of quadrilateral ABCD.

In Fig. 15.75, PQRS is a square and T and U are respectively, the mid-points of PS and QR. Find the area of Δ OTS if PQ=8 cm

Compute the area of trapezium PQRS in Fig. 15.76.

In Fig. 15.77, ∠AOB=90°, AC=BC, OA=12 cm and OC=6,5 cm. Find the area of Δ AOB.

In Fig. 15.78, ABCD is a trapezium in which AB=7 cm, AD=BC=5 cm, DC=x cm, and distance between AB and DC is 4 cm. Find the value of x and area of trapezium ABCD.

In Fig. 15.79, OCDE is a rectangle inscribed in a quadrilateral of a circle of radium 10 cm. If OE=2 , find the area of the rectangle.

In Fig. 15.80, ABCD is a trapezium in which AB||DC. Prove that ar(Δ AOD) = ar(Δ BOC).

In Fig. 15.81, ABCD and CDEF are parallelograms. Prove that

ar(Δ ADE) = ar(Δ BCF).

Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that:

ar(Δ APB) × ar(Δ CPD) = ar(Δ APD) × ar(Δ BPC).

In Fig. 15.82, ABC and ABD are two triangles on the base AB. If line segment CD is bisected by AB at O, Show that ar(Δ ABC) = ar(Δ ABD).

If P is any point in the interior of a parallelogram ABCD, then prove that area of the triangle APB is less than half the area of parallelogram.

If AD is a median of a triangle ABC, then prove that triangles ADB and ADC are equal in area. If G is the mid-point of median AD, prove that ar(Δ BGC) = 2ar(Δ AGC).

A point D is taken on the side BC of a Δ ABC such that BD = 2DC. Prove that

ar(Δ ABD) = 2ar(Δ ADC)

ABCD is a parallelogram whose diagonals intersect at O. If P is any point on BO, prove that

(i) ar(Δ ADO) = ar(Δ CDO)

(ii) ar(Δ ABP) = ar(Δ CBP)

ABCD is a parallelogram in which BC is produced to E such that CE=BC. AE intersects CD at F.

(i) Prove that ar(Δ ADF) = ar(Δ ECF)

(ii) If the area of Δ DFB=3 cm², find the area of ||^gm ABCD.

ABCD is a parallelogram whose diagonals AC and BD intersect at O. A line through O intersects AB at P and DC at Q. Prove that

ar(Δ POA) = ar(Δ QOC)

ABCD is a parallelogram. E is a point on BA such that BE = 2 EA and F is a point on DC, such that DF=2FC. Prove that AE CF is a parallelogram whose area is one third of the area of parallelogram ABCD.

In a Δ ABC, P and Q are respectively the mid-point of AB and BC and R is the mid-point of AP. Prove that:

(i) ar(Δ PBQ) = ar(Δ ARC)

(ii) ar(Δ PQR) = ar(Δ ARC)

(iii) ar(Δ RQC) = ar(Δ ABC)

ABCD is a parallelogram, G is the point on AB such that AG = 2GB, E is a point of DC such that CE = 2DE and F is the point of BC such that BF=2FC. Prove that:

(i) ar(Δ ADGE) = ar(Δ GBCE)

(ii) ar(Δ EGB) = ar(Δ ABCD)

(iii) ar(Δ EFC) = ar(Δ EBF)

(iv) ar(Δ EBG) = ar(Δ EFC)

In Fig. 15.83, CD||AE and CY||BA.

(i) Name a triangle equal in area of Δ CBX

(ii) Prove that ar(Δ ZDE) = ar(Δ CZA)

(iii) Prove that ar(Δ BCYZ) = ar(Δ EDZ)

In Fig. 15.84, PSDA is a parallelogram in which PQ=QR=RS and AP||BQ||CR. Prove that

ar(Δ PQE) = ar(Δ CFD)

In Fig. 15.85, ABCD is a trapezium in which AB||DC and DC=40 cm and AB=60 cm. If X and Y are, respectively, the mid points of AD and BC, Prove that

(i) XY =50 cm (ii) DCYX is a trapezium

(iii) ar(trap. DCYX) = ar(trap. XYBA)

In Fig. 15.86, ABC and BDE are two equilateral triangles such that D is the mid-point of BC. AE intersects BC in F. Prove that

(i) ar(Δ BDE) = ar(Δ ABC)

(ii) ar(Δ BDE) = ar(Δ BAE)

(iii) ar(Δ BFE) = ar(Δ AFD)

(iv) ar(Δ ABC) = ar(Δ BEC)

(v) ar(Δ FED) = ar(Δ AFC)

(vi) ar(Δ BFE) = 2ar(Δ EFD)

D is the mid-point of side BC of Δ ABC and E is the mid-point of BD. If O is the mid-point of AE, prove that ar(ΔBOE) = ar(Δ ABC)

In Fig. 15.87, X and Y are the mid-point of AC and AB respectively, QP||BC and CYQ and BXP are straight lines. Prove that:

ar(Δ ABP) = ar(Δ ACQ).

In Fig. 15.88, ABCD and AEFD are two parallelograms. Prove that:

(i) PE=FQ

(ii) ar(ΔAPE):ar(Δ PFA)= ar(Δ QFD)=ar(Δ PFD)

(iii) ar(Δ PEA) = ar(Δ QFD)

In Fig. 15.89, ABCD is a ||^gm. O is any point on AC. PQ||AB and LM||AD. Prove that:

ar(||^gm DLOP) = ar(||^gm BMOQ)

In a Δ ABC, if L and M are points on AB and AC respectively such that LM||BC. Prove that:

(i) ar(Δ LCM) = ar(Δ LBM)

(ii) ar(Δ LBC) = ar(Δ MBC)

(iii) ar(Δ ABM) = ar(Δ ACL)

(iv) ar(Δ LOB) = ar(Δ MOC)

In Fig. 15.90, D and E are two points on BC such that BD=DE=EC. Show that ar(Δ ABD) = ar(Δ ADE) =ar(Δ AEC)

In Fig. 15.91, ABC is a right triangle right angled at A, BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment AX⊥ DE meets BC at Y. Show that:

(i) Δ MBC ≅ ABD (ii) ar(BYXD) =2ar(Δ MBC)

(iii) ar(BYXD) = ar(ABMN)

(iv) Δ FCB ≅ Δ ACE

(v) ar(CYXE) = 2ar(Δ FCB)

(vi) ar(CYXE) = ar(ACFG)

(vii) ar(BCED) = ar(ABMN)+ ar(ACFG)

If ABC and BDE are two equilateral triangles such that D is the mid-point of BC, then find

In Fig. 15.104, ABCD is a rectangle in which CD = 6 cm, AD = 8 cm. Find the area of parallelogram CDEF.

In Fig. 15.104, find the area of ΔGEF.

In Fig. 15.105, ABCD is a rectangle with sides AB = 10 cm and AD = 5 cm. Find the area of ΔEFG.

PQRS is a rectangle inscribed in a quadrant of a circle of radius 13 cm. A is any point on PQ. If PS = 5 cm, then find

In square AB2CD, P and Q are mid-point of AB and CD respectively. If AB = 8 cm and PQ and BD intersect at O, then find area of ΔOPB.

ABC is a triangle in which D is the mid-point of BC. E and F are mid-points of DC and AE respectively. If area of ΔABC is 16 cm², find the area of ΔDEF.

PQRS is a trapezium having PS and QR as parallel sides. A is any point on PQ and B is a point on SR such that AB||QR. If area of ΔPBQ is 17 cm², find the area of ΔASR.

ABCD is a parallelogram. P is the mid-point of AB. BD and CP intersect at Q such that CO:OP = 3:1. If Find the area of parallelogram ABCD.

P is any point on base BC of ΔABC and D is the mid-point of BC. DE is drawn parallel to PA to meet AC at E. If then find area of ΔEPC.

The opposite sides of a quadrilateral have

Two consecutive sides of a quadrilateral have

PQRS is a quadrilateral. PR and QS intersect each other at O. In which of the following cases, PQRS is a parallelogram?

Which of the following quadrilateral is not a rhombus?

Diagonals necessarily bisect opposite angles in a

The two diagonals are equal in a

We get a rhombus by joining the mid-points of the sides of a

The bisectors of any two adjacent angles of a parallelogram intersect at

The bisectors of the angle of a parallelogram enclose a

The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a

The figure formed by joining the mid-points of the adjacent sides of a rectangle is a

The figure formed by joining the mid-points of the adjacent sides of a rhombus is a

The figure formed by joining the mid-points of the adjacent sides of a square is a

The figure formed by joining the mid-points of the adjacent sides of a parallelogram is a

If one side of a parallelogram is 24° less than twice the smallest angle, then the measure of the largest angle of the parallelogram is

In a parallelogram ABCD, if ∠DAB=75° and ∠DBC=60°, then ∠BDC=

ABCD is a parallelogram and E and F are the centroids of triangles ABD and BCD respectively, then EF=

ABCD is a parallelogram M is the mid-point of BD and BM bisects ∠B. Then, ∠AMB=

ABCD is a parallelogram and E is the mid-point of BC. DE and AB when produced meet at F. Then, AF =

If an angle of a parallelogram is two-third of its adjacent angle, the smallest angle of the parallelogram is

If the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?

In a quadrilateral ABCD, ∠A+∠C is 2 times ∠B+∠D. If ∠A= 140° and f∠D = 60°, then ∠B=

If the diagonals of a rhombus are 18 cm and 24 cm respectively, then its side is equal to

The diagonals AC and BD of a rectangle ABCD intersect each other at P. If ∠ABD =50°, then ∠DPC =

ABCD is a parallelogram in which diagonal AC bisects ∠BAD. If ∠BAC =35°, then ∠ABC =

In a rhombus ABCD, if ∠ACB =40°, then ∠ADB =

In Δ ABC, ∠A =30°, ∠B=40° and ∠C=110°. The angles of the triangle formed by joining the mid-points of the sides of this triangle are

The diagonals of a parallelogram ABCD intersect ay O. If ∠BOC =90° and ∠BDC=50°, then ∠OAB =

ABCD is a trapezium in which AB||DC. M and N are the mid-points of AD and BC respectively, If AB=12cm, MN=14 cm, then CD=

Diagonals of a quadrilateral ABCD bisect each other. If ∠A =45°, then ∠B =

P is the mid-point of side BC of a parallelogram ABCD such that ∠BAP =∠DAP. If AD=10 cm, then CD=

In Δ ABC, E is the mid-point of median AD such that BE produced meets AC at F. If AC = 10.5 cm, then AF =