Selina solutions for Concise Mathematics [English] Class 9 ICSE chapter 15 - Area Theorems [Proof and Use] [Latest edition]

In the given figure, BD : DC = 3 : 5, then area of ΔABD : area of ΔACD is:

A median of a triangle divides it into two ______.

In the given figure, AB is parallel to DС and AB ≠ DC, the area of ΔAOD is equal to area of triangle:

In the given figure, AF//BE and PQ//RS, FC and ED are perpendiculars to RS. The area of parallelogram ABEF is equal to:

In the given figure, D is mid-point of side BC, the area of triangle BEA is equal to area of triangle:

In the given figure, if the area of triangle ADE is 60 cm², state, given reason, the area of :
(i) Parallelogram ABED;
(ii) Rectangle ABCF;
(iii) Triangle ABE.

The given figure shows a rectangle ABDC and a parallelogram ABEF; drawn on opposite sides of AB.
Prove that: 
(i) Quadrilateral CDEF is a parallelogram;
(ii) Area of the quad. CDEF
= Area of rect. ABDC + Area of // gm. ABEF.

In the given figure, diagonals PR and QS of the parallelogram PQRS intersect at point O and LM is parallel to PS. Show that:

(i) 2 Area (POS) = Area (// gm PMLS)
(ii) Area (POS) + Area (QOR) = Area (// gm PQRS)
(iii) Area (POS) + Area (QOR) = Area (POQ) + Area (SOR).

In parallelogram ABCD, P is a point on side AB and Q is a point on side BC.
Prove that:
(i) ΔCPD and ΔAQD are equal in the area.
(ii) Area (ΔAQD) = Area (ΔAPD) + Area (ΔCPB)

In the given figure, M and N are the mid-points of the sides DC and AB respectively of the parallelogram ABCD.

If the area of parallelogram ABCD is 48 cm²;
(i) State the area of the triangle BEC.
(ii) Name the parallelogram which is equal in area to the triangle BEC.

In the following figure, CE is drawn parallel to diagonals DB of the quadrilateral ABCD which meets AB produced at point E.
Prove that ΔADE and quadrilateral ABCD are equal in area.

ABCD is a parallelogram a line through A cuts DC at point P and BC produced at Q. Prove that triangle BCP is equal in area to triangle DPQ.

The given figure shows a pentagon ABCDE. EG drawn parallel to DA meets BA produced at G and CF draw parallel to DB meets AB produced at F.

Prove that the area of pentagon ABCDE is equal to the area of triangle GDF.

In the given figure, AP is parallel to BC, BP is parallel to CQ.
Prove that the area of triangles ABC and BQP are equal.

In the figure given alongside, squares ABDE and AFGC are drawn on the side AB and the hypotenuse AC of the right triangle ABC.

If BH is perpendicular to FG

prove that:

1. ΔEAC ≅ ΔBAF
2. Area of the square ABDE
3. Area of the rectangle ARHF.

In the following figure, DE is parallel to BC.
Show that: 
(i) Area ( ΔADC ) = Area( ΔAEB ).
(ii) Area ( ΔBOD ) = Area( ΔCOE ).

Show that:

A diagonal divides a parallelogram into two triangles of equal area.

Show that:

The ratio of the areas of two triangles of the same height is equal to the ratio of their bases.

Show that:
The ratio of the areas of two triangles on the same base is equal to the ratio of their heights.

In the given figure; AD is median of ΔABC and E is any point on median AD.
Prove that Area (ΔABE) = Area (ΔACE).

In the figure, if E is the mid-point of median AD, then prove that:

Area (ΔABE) = `1/4` Area (ΔABC).

ABCD is a parallelogram. P and Q are the mid-points of sides AB and AD respectively.
Prove that area of triangle APQ = `1/8` of the area of parallelogram ABCD.

The base BC of triangle ABC is divided at D so that BD = `1/2`DC.
Prove that area of ΔABD = `1/3` of the area of ΔABC.

In a parallelogram ABCD, point P lies in DC such that DP: PC = 3:2. If the area of ΔDPB = 30 sq. cm.
find the area of the parallelogram ABCD.

The median of a triangle divides it into two ______.

In the following figure, the area of parallelogram ABCD is ______.

ABCD is a quadrilateral whose diagonals intersect each other at point O. The diagonal AC bisects diagonal BD. Then area of quadrilateral ABCD is ______.

Two parallelograms ABCD and ABEF are equal in area, they lie between the same parallel lines:

ABCD is a trapezium with parallel sides AB = a cm and DC = b cm (Figure). E and F are the mid-points of the non-parallel sides. The ratio of ar (ABFE) and ar (EFCD) is ______.

Statement (1): ABCD is a quadrilateral whose diagonal AC divides it into two parts, equal in area.

Statement (2): It is not necessary that the quadrilateral ABCD is a rectangle or a parallelogram or rhombus.

Assertion (A): PQRS a parallelogram whose area is 180 cm² and A is any point on the diagonal PR. The area of triangle ASR = 45 cm².

Reason (R): A is not the mid-point of diagonal PR.

Assertion (A): ABCD is a square. E is mid-point of side AB and F is mid-point of side DC. If DA = 16 cm, the area of triangle COF is 32 cm².

Reason (R): EF is ⊥ to DC and OF = `1/2` EF = `1/2` DA = 8 cm. Area of COF = `1/2` × CF × OF.

ABCD and BCFE are parallelograms. If area of triangle EBC = 480 cm²; AB = 30 cm and BC = 40 cm.

Calculate : 
(i) Area of parallelogram ABCD;
(ii) Area of the parallelogram BCFE;
(iii) Length of altitude from A on CD;
(iv) Area of triangle ECF.

In the given figure, D is mid-point of side AB of ΔABC and BDEC is a parallelogram.

Prove that: Area of ABC = Area of // gm BDEC.

In the following, AC // PS // QR and PQ // DB // SR.

Prove that: Area of quadrilateral PQRS = 2 x Area of the quad. ABCD.

ABCD is a trapezium with AB // DC. A line parallel to AC intersects AB at point M and BC at point N.
Prove that: area of Δ ADM = area of Δ ACN.

In the given figure, AD // BE // CF.
Prove that area (ΔAEC) = area (ΔDBF)

In the given figure, ABCD is a parallelogram; BC is produced to point X.
Prove that: area ( Δ ABX ) = area (`square`ACXD )

The given figure shows the parallelograms ABCD and APQR.
Show that these parallelograms are equal in the area.
[ Join B and R ]

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

If ar.(∆DFB) = 30 cm²; find the area of parallelogram.

The following figure shows a triangle ABC in which P, Q, and R are mid-points of sides AB, BC and CA respectively. S is mid-point of PQ:
Prove that: ar. ( ∆ ABC ) = 8 × ar. ( ∆ QSB )

In the given figure, the diagonals AC and BD intersect at point O. If OB = OD and AB//DC,
show that:
(i) Area (Δ DOC) = Area (Δ AOB).
(ii) Area (Δ DCB) = Area (Δ ACB).
(iii) ABCD is a parallelogram.

The given figure shows a parallelogram ABCD with area 324 sq. cm. P is a point in AB such that AP: PB = 1:2
Find The area of Δ APD.

In ΔABC, E and F are mid-points of sides AB and AC respectively. If BF and CE intersect each other at point O,
prove that the ΔOBC and quadrilateral AEOF are equal in area.

In parallelogram ABCD, P is the mid-point of AB. CP and BD intersect each other at point O. If the area of ΔPOB = 40 cm², and OP: OC = 1:2, find:
(i) Areas of ΔBOC and ΔPBC
(ii) Areas of ΔABC and parallelogram ABCD.

The medians of a triangle ABC intersect each other at point G. If one of its medians is AD,
prove that:
(i) Area ( ΔABD ) = 3 x Area ( ΔBGD )
(ii) Area ( ΔACD ) = 3 x Area ( ΔCGD )
(iii) Area ( ΔBGC ) = `1/3` x Area ( ΔABC ).

The perimeter of a triangle ABC is 37 cm and the ratio between the lengths of its altitudes be 6: 5: 4. Find the lengths of its sides.
Let the sides be x cm, y cm, and (37 - x - y) cm. Also, let the lengths of altitudes be 6a cm, 5a cm, and 4a cm.

In parallelogram ABCD, E is a point in AB and DE meets diagonal AC at point F. If DF: FE = 5:3 and area of  ΔADF is 60 cm²; find
(i) area of ΔADE.
(ii) if AE: EB = 4:5, find the area of  ΔADB.
(iii) also, find the area of parallelogram ABCD.

In the following figure, BD is parallel to CA, E is mid-point of CA and BD = `1/2`CA
Prove that: ar. ( ΔABC ) = 2 x ar.( ΔDBC )

In the following figure, OAB is a triangle and AB || DC.

If the area of ∆ CAD = 140 cm² and the area of ∆ ODC = 172 cm²,
find : (i) the area of ∆ DBC
(ii) the area of ∆ OAC
(iii) the area of ∆ ODB.

E, F, G, and H are the midpoints of the sides of a parallelogram ABCD.
Show that the area of quadrilateral EFGH is half of the area of parallelogram ABCD.