Frank solutions for Mathematics [English] Class 9 ICSE chapter 21 - Areas Theorems on Parallelograms [Latest edition]

ABCD is a parallelogram having an area of 60cm2. P is a point on CD. Calculate the area of ΔAPB.

PQRS is a rectangle in which PQ = 12cm and PS = 8cm. Calculate the area of ΔPRS.

In the figure, PT is parallel to SR. QTSR is a parallelogram and PQSR is a rectangle. If the area of ΔQTS is 60cm², find:
(i) the area o || gm QTSR
(ii) the area of the rectangle PQRS
(iii) the area of the triangle PQS.

In the given figure area of ∥ gm PQRS is 30 cm². Find the height of ∥ gm PQFE if PQ = 6 cm.

In the given figure, PQRS is a ∥ gm. A straight line through P cuts SR at point T and QR produced at N. Prove that area of triangle QTR is equal to the area of triangle STN.

In the given figure, ST ∥ PR. Prove that: area of quadrilateral PQRS = area of ΔPQT.

In the figure, ABCD is a parallelogram and APD is an equilateral triangle of side 80cm, Calculate the area of parallelogram ABCD.

In the figure, if the area of ||gm PQRS is 84cm²; find the area of 
(i) || gm PQMN
(ii) ΔPQS
(iii) ΔPQN

In the figure, PQR is a straight line. SQ is parallel to Tp. Prove that the quadrilateral PQST is equal in area to the ΔPSR.

In the given figure, if AB ∥ DC ∥ FG and AE is a straight line. Also, AD ∥ FC. Prove that: area of ∥ gm ABCD = area of ∥ gm BFGE.

In the given figure, the perimeter of parallelogram PQRS is 42 cm. Find the lengths of PQ and PS.

In the given figure, PT ∥ QR and QT ∥ RS. Show that: area of ΔPQR = area of ΔTQS.

*Question modified

In the given figure, ΔPQR is right-angled at P. PABQ and QRST are squares on the side PQ and hypotenuse QR. If PN ⊥ TS, show that:
(a) ΔQRB ≅ ΔPQT
(b) Area of square PABQ = area of rectangle QTNM.

In the figure, AE = BE. Prove that the area of triangle ACE is equal in area to the parallelogram ABCD.

The diagonals of a parallelogram ABCD intersect at O. A line through O meets AB in P and CD in Q. Show that
(a) Area of APQD = `(1)/(2)` area of || gm ABCD

(b) Area of APQD = Area of BPQC

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

AD is a median of a ΔABC.P is any point on AD. Show that the area of ΔABP is equal to the area of ΔACP.

In the given figure AF = BF and DCBF is a parallelogram. If the area of ΔABC is 30 square units, find the area of the parallelogram DCBF.

Prove that the diagonals of a parallelogram divide it into four triangles of equal area.

The diagonals AC and BC of a quadrilateral ABCD intersect at O. Prove that if BO = OD, then areas of ΔABC an ΔADC area equal.

Prove that the area of a rhombus is equal to half the rectangle contained by its diagonals.

PQRS is a parallelogram and O is any point in its interior. Prove that: area(ΔPOQ) + area(ΔROS) - area(ΔQOR) + area(ΔSOP) = `(1)/(2)`area(|| gm PQRS)

A quadrilateral ABCD is such that diagonals BD divides its area into two equal parts. Prove that BD bisects AC.

In the given figure, BC ∥ DE.
(a) If area of ΔADC is 20 sq. units, find the area of ΔAEB.
(b) If the area of ΔBFD is 8 square units, find the area of ΔCEF

ΔPQR and ΔSQR are on the same base QR with P and S on opposite sides of line QR, such that area of ΔPQR is equal to the area of ΔSQR. Show that QR bisects PS.

If the medians of a ΔABBC intersect at G, show that ar(ΔAGB) = ar(ΔAGC) = ar(ΔBGC) = `(1)/(3)"ar(ΔABC)"`.

In ΔABC, the mid-points of AB, BC and AC are P, Q and R respectively. Prove that BQRP is a parallelogram and that its area is half of ΔABC.

In the given figure, PQ ∥ SR ∥ MN, PS ∥ QM and SM ∥ PN. Prove that: ar. (SMNT) = ar. (PQRS).

In ΔPQR, PS is a median. T is the mid-point of SR and M is the mid-point of PT. Prove that: ΔPMR = `(1)/(8)Δ"PQR"`.

In the figure, ABCD is a parallelogram and CP is parallel to DB. Prove that: Area of OBPC = `(3)/(4)"area of ABCD"`

The medians QM and RN of ΔPQR intersect at O. Prove that: area of ΔROQ = area of quadrilateral PMON.

In the given figure, ABC is a triangle and AD is the median.

If E is any point on the median AD. Show that: Area of ΔABE = Area of ΔACE.

In the given figure, ABC is a triangle and AD is the median.

If E is the midpoint of the median AD, prove that: Area of ΔABC = 4 × Area of ΔABE

In a parallelogram PQRS, M and N are the midpoints of the sides PQ and PS respectively. If area of ΔPMN is 20 square units, find the area of the parallelogram PQRS.

In a parallelogram PQRS, T is any point on the diagonal PR. If the area of ΔPTQ is 18 square units find the area of ΔPTS.

ABCD is a quadrilateral in which diagonals AC and BD intersect at a point O. Prove that: areaΔAOD + areaΔBOC + areaΔABO + areaΔCDO.

In the given figure, AB ∥ SQ ∥ DC and AD ∥ PR ∥ BC. If the area of quadrilateral ABCD is 24 square units, find the area of quadrilateral PQRS.