B Nirmala Shastry solutions for मॅथेमॅटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 13 - Theorems on Area [Latest edition]

Find the area of parallelogram ABCD, if AD = 15 cm and P is a point on BC such that AP = 9 cm. Also find the area of parallelogram ABQR, where Q and R are points on CD produced.

M is a point on side PQ of rectangle PQRS. SR is produced to a point N and MSRT is a parallelogram. PQ = 8.5 cm, PS = 6 cm. Find the area of

1. parallelogram MSRT
2. ΔMNT

ABCD is a rectangle, AB = 15 cm, AC = 17 cm. Find the area of parallelogram PQCA.

ABCD is a square of side 10 cm. Find the area of parallelogram BDPQ.

The perimeter of parallelogram ABCD is 42 cm. If AP = 3 cm and AQ = 4 cm, calculate the sides of parallelogram ABCD.

P is any point on the side AB of parallelogram ABCD. Prove that area (ΔAPD) + area (ΔPBC) = `1/2` area (|| gm ABCD).

PQRS is a parallelogram. A is any point on SR. PA is produced to meet QR produced at B. Prove that area (ΔQAR) = area (ΔSAB).

ABCD and PBCQ are parallelograms. Area of ΔPBC = 8 cm². Calculate the area of parallelograms ABCD, PBCQ and ΔPCQ.

In the given figure, AC || BH and AD || EF. Show area (ΔAHF) = area (pentagon ABCDE). [Hint: Join BC and DE.]

[Hint: Join BC and DE.]

ABCD is a rectangle, AB = 4 cm, BC = 10 cm. Find the area of ΔRDC and || gm PQCD.

ABCD and ABPQ are two parallelograms. Side AQ and BC are produced to meet at R. Show that area (ΔBCD) = area (ΔBPR).

In the given figure, TQ || SR and PQ || TS. Prove that area (ΔPTS) = area (ΔTRQ).

[Hint: Join QS.]

ABCD and APDR are parallelograms. P is the mid-point of BC and Q is on CD such that DQ : QC = 2 : 1. Area (ΔPQC) = 20 cm². Find the areas of 

1. ΔPDC
2. ΔABP
3. || gm APDR

ABCD and BEFG are parallelograms and CE || AG. Prove that Area (|| gm ABCD) = Area (|| gm BEFG) 

[Hint: Prove Area (ΔABC) = Area (ΔGBE)]

ABCD is a parallelogram, BC is produced to Q so that ∠DQC = 90°. AP ⊥ DC. If AP = 2.5 cm, BC = 4 cm and DQ = 5 cm, find the length of AB.

[Hint: Area of || gm = CD × AP = AD × DQ]

ABCD is a parallelogram and AF || DE. Prove that area (|| gm DEFH) = area (|| gm ABCD).

[Hint: Each || gm is equal in area to || gm ADEG]

In ΔABC, M is the mid-point of BC, L is a point on AB such that AL = 2LB. Find the area of ΔALM if area of ΔABC = 72 cm².

ABCD is a parallelogram, prove that 

1. Area (ΔABC) = Area (ΔPAD)
2. Area (ΔPAB) = Area (quadrilateral ACPD)

D is the mid-point of AC. E is on BC such that BE = `1/3` BC. Area of ΔABC = 48 cm². Find the area of ΔADB and ΔBDE.

Area of ΔPCD = 10 cm². Find the area of ΔADQ and parallelogram ABCD.

P is a point on side AB of parallelogram ABCD. Which of the following is true?

1. Area of ΔAPD = `1/4` area || gm ABCD
2. Area of ΔPDC = `1/2` area || gm ABCD
3. Area of ΔAPD + area of ΔPBC = `1/2` parallelogram ABCD

In || gm ABCD, M is a point on AB, ∠DMC = 90°. DM = 15 cm, DC = 17 cm. ∴  Area of || gm ABCD is:

PQRS is a rectangle and AQRB is a || gm, SR = 9 cm, PS = 12 cm

Area of ΔCQR =

PQRS is a rectangle and AQRB is a || gm, SR = 9 cm, PS = 12 cm

Area of || gm AQRB =

Area of ΔPMQ = 72 cm², Area of || gm LMNO =

ABCD is a rectangle. PACQ is a || gm. AB = 12 cm, AC = 15 cm. Area of || gm PACQ =

Area of || gm ABCD =

AM is the median of ΔABC. Area of ΔABC = 140 cm². AN : NC = 3 : 2. Area of ΔMNC =

P and Q are mid points of AB and AC of ΔABC. Which of the following is not true?

In ΔABC, P, M and N are mid points of sides AB, BC and AC. If area of ΔABC = 120 cm², Area of || gm APMN =

ABCD is a rectangle, DBCE is a || gm. Area of || gm DBCE =

Assertion: If area of ΔAOD = 18 cm² then area of parallelogram ABCD = 72 cm².

Reason: The diagonals of a parallelogram bisect each other.

Assertion: Area of ΔPDC = 160 cm² and area of parallelogram ABCD = 320 m².

Reason: Area of a right angled Δ = `1/2` product of sides containing right angle.

Assertion: A median of a triangle divides it into 2 triangles of equal area.

Reason: The two triangles so formed are congruent.

Assertion: AB || DC in the figure. Area of ΔAOD = Area of ΔBOC.

Reason: Two Δs on the same base and between the same parallel lines have equal area.

ABCD is a trapezium in which AD || BC. M is the mid-point of CD. Through M, PQ is drawn || to AB with P on AD produced and Q on BC.

1. Show that ΔDPM ≅ ΔCQM
2. Prove that area of ΔABM = `1/2` × area of trapezium ABCD.

The area of ΔABC = 18 cm². Find the distance between the || lines AD and BC if BC = 8 cm. Calculate the area of || gm BCDE.

ABCD is a rectangle and BCED is a parallelogram. If BC = 12 cm and CE = 15 cm, find the area of

1. Parallelogram BCED
2. ΔCDE

In ΔPQR, PM and QN are medians. Area of ΔGQM = 15 cm². Find the area of ΔPQR.

ABCD is a trapezium with AB with AB || DC and diagonals AC and BD intersect at O. Prove that area of ΔAOD = area of ΔBOC.

Area of ΔABD = 30 cm². Find the area of ΔADC, if BD = `5/8` BC.

AM is the median of ΔABC. N is on AC such that AN : NC = 3 : 2. If area of ΔMNC = 12 cm², find the area of ΔABC.

In ΔABC, PQ || BC. Prove that

1. Area (ΔPBC) = Area (ΔQBC)
2. Area (ΔPOB) = Area (ΔQOC)

In adjoining figure, ABCD is a parallelogram, AQ = QB. CB and DQ are produced to meet at P. Prove that

1. Area (ΔAQD) = Area (ΔAQP)
2. Area (ΔDCP) = Area (|| gm ABCD)

In the given figure, PQRS and PXYZ are parallelograms. Prove that they are of equal area.

[Hint: Join XQ. Area (ΔXPQ) = `1/2` of each || gm]

In the given figure, PQRS and PXYZ are two parallelograms of equal area. Prove that SX || YR.