Nootan solutions for माठेमटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 13 - Theorems on Area [Latest edition]

In the adjoining figure; if the area of parallelogram ABCD is 90 cm², find :

1. area of parallelogram ABEF.
2. area of ΔABE.

In the adjoining figure; AB = 16 cm, DM = 4 cm and BC = 8 cm. Find the length of DN.

In the adjoining figure, BD = 2CD. If area of ΔABC = 36 cm², find the area of ΔABD.

In the adjoining figure, ABCD is a rectangle in which AB = 10 cm, AD = 6 cm. If AP || BQ, find :

1. ar (|| gm ABQP) 
2. ar (ΔАВР)

The area of a parallelogram is 150 cm². The ratio of its base and corresponding altitude is 3 : 2. Find the base and corresponding altitude.

In the adjoining figure; AN = NB and DM = MC in the parallelogram ABCD. If area (ABCD) = 44 cm²,

1. find the area of ΔВСЕ. 
2. find the area of parallelogram BNMC.

In the adjoining figure; ABCD is a parallelogram. If AP = BP and CP meets the diagonal BD at Q and area of ΔBPQ = 20 cm², find:

1. PQ : QC.
2. Area of ΔPBC.
3. Area of parallelogram ABCD.

In the adjoining figure; ABCD is a parallelogram whose area is 324 cm². If P is a point on AB such that AP : PB = 1 : 2, find:

1. Area of ΔAPD. 
2. QP : QD.

In a parallelogram ABCD, P is a point on DC such that DP : PC = 2 : 3. If area of ΔDPB = 40 cm², find the area of ◻ABCD.

In the adjoining figure; BD = DC and AE = ED. Prove that area of ΔACE = `1/4` × area of ΔАВС.

In the adjoining figure, D is the mid-point of BC and E be any point on AD.

Prove that : 

1. area (ΔEBD) = area (ΔECD).
2. area (ΔABE) = area (ΔACE).

In a parallelogram ABCD, M and N are the points on AB and BC respectively.

Prove that:

1. ΔCMD and ΔAND are equal in area. 
2. area (ΔAND) = area (ΔAMD) + area (ΔCMB).

In a trapezium ABCD, AB || DC and the diagonals AC and BD intersect at point ‘O’. Prove that area (ΔАOD) = area (ΔBОС).

In the adjoining figure, O is the mid-point of diagonal AC of a quadrilateral ABCD. Prove that area (◻ABOD) = area (◻BODC).

In the adjoining figure, AD = BD and ◻BDEC is a parallelogram. Prove that area (ΔABC) = area (◻BDEC).

In the adjoining figure, PQ || BC.

Prove that :

1. area (ΔABQ) = area (ΔАСР). 
2. area (ΔΒΟP) = area (ΔCOQ).

In the adjoining figure, ABCD is a parallelogram. A line through A intersects DC at P and meets BC produced at Q.

Prove that : area (ΔBCP) = area (ΔDPQ).

In the adjoining figure, D is the mid-point of AB. P is any point on BC and CQ || PD meets AB at Q. 

Prove that : area (ΔBPQ) = `1/2` × area (ΔABC)

In a trapezium ABCD, AB || DC. A line parallel to AC cuts AB at P and BC at Q. Prove that area (ΔАPD) = area (ΔACQ).

In the adjoining figure, ΔABC is a right-angled triangle, right-angled at B; ABDE and ACGF are the squares. If BH is perpendicular to FG.

Prove that:

1. area (ΔΕAC) = area (ΔΒΑF).
2. area (◻ABDE) = area (◻AIHF).

The diagonals AC and BD of a parallelogram ABCD intersect at ‘O’. A straight line through ‘O’ meets AB at P and DC at Q. 

Prove that: area (◻APQD) = `1/2` × area (◻ABCD).

In the adjoining figure, ABCDE is a pentagon in which EG is drawn parallel to DA meets BA produced at G and CF is drawn parallel to DB meets AB produced at F. Prove that area of ABCDE = area of GDF.

In the adjoining figure, BD = CD and P is a point on AC such that area (ΔAPD) : area (ΔАBD) = 2 : 3. 

Find:

1. AP : PC. 
2. area (ΔDCP) : area (ΔАВС).

In the adjoining figure, ABCD and BEFG are two parallelograms.

Prove that : area (◻ABCD) = area (◻BEFG).

ABCD 8 is a parallelogram. E and F are the mid-points of the sides AB and AD respectively. Prove that area (ΔAEF) = `1/8` × area (◻ABCD).

In ΔAВC; E and F are the mid-points of sides AB and AC respectively. If FB and CE intersect at point ‘O’, prove that area (ΔOBC) = area (◻AEOF).

In the adjoining figure, ABCD and ABFE are two parallelograms.

Prove that: area (ABCD) + area (ABFE) = area (EFCD).

In the adjoining figure, PQ || BC, BE || CA and CF || BA. Prove that area of ΔABE = area of ΔACF.

Two parallelograms are on equal bases and between the same parallel lines. The ratio of their areas is ______.

In the adjoining figure, a parallelogram ABCD is shown. If DM ⊥ AB, DN ⊥ BC, then area of ΔABD is equal to:

In the adjoining figure, a trapezium ABCD is shown in which AB || DC then, ar (ΔAOD) is equal to:

A triangle and a rectangle are on the same base and between the same parallel lines. The ratio of their areas is ______.

In the adjoining figure, ABCD is a parallelogram and M is the mid-point of AВ. If ar (|| gm ABCD) = 40 cm², then ar (ΔADM) is equal to:

The median of a triangle divides it into two ______.

In ΔАВC; the mid-points of the sides BC, CA and AB are D, E and F respectively. The ratio of ar (ΔDEF) and ar (ΔABC) is ______.

In the adjoining figure, a trapezium ABCD is shown in which AB || DC, AB = x and DC = y. If E and F are the mid-points of the sides AD and BC respectively, then ar (ABFE) : ar (EFCD) is:

In the adjoining figure, a parallelogram ABCD is shown in which P divides AB in the ratio 1 : 2. If the area of the parallelogram ABCD is 60 cm², then area (BPDC) is:

In the adjoining figure, BM = 2CM. If ar (ΔАBC) = 30 cm², then ar (ΔABM) is: