NCERT Class 9 Solutions: Areas of Parallelograms and Triangles (Chapter 9) Exercise 9.1

Trapezoid is a quadrilateral with one pair of parallel side two pairs of base angles. ABCD is a trapezoid

Trapezoid

Trapezoid is a quadrilateral with one pair of parallel side two pairs of base angles. ABCD is a trapezoid

A parallelogram and its properties

About Parallellogram

A parallelogram and its properties

Q-1 Which of the following figure lie on the same base and between the same parallel. In such a case, write the common base and the two parallels.

Segment f Segment f: Segment [A, B] Segment g Segment g: Segment [A, D] Segment h Segment h: Segment [B, C] Segment j Segment j: Segment [D, P] Segment i Segment i: Segment [P, C] Segment P_1 Segment P_1: Segment [D, C] Point A A = (1.24, 1.04) Point A A = (1.24, 1.04) Point A A = (1.24, 1.04) Point B B = (3.86, 1.04) Point B B = (3.86, 1.04) Point B B = (3.86, 1.04) Point D D = (0.74, -0.68) Point D D = (0.74, -0.68) Point D D = (0.74, -0.68) Point C C = (4.2, -0.64) Point C C = (4.2, -0.64) Point C C = (4.2, -0.64) Point P Point P: Point on f Point P Point P: Point on f Point P Point P: Point on f (i) text1 = "(i)"

Fig 1: Parallelogram, Trapezium and Triangles

Figure with different types of parallelogram, trapezium and triangles

Segment f Segment f: Segment [P, Q] Segment g Segment g: Segment [P, S] Segment h Segment h: Segment [Q, R] Segment i Segment i: Segment [S, R] Segment j Segment j: Segment [M, N] Segment k Segment k: Segment [M, S] Segment l Segment l: Segment [N, R] Point P P = (2.06, 1.92) Point P P = (2.06, 1.92) Point P P = (2.06, 1.92) Point Q Q = (5.22, 1.9) Point Q Q = (5.22, 1.9) Point Q Q = (5.22, 1.9) Point S S = (0.74, 0.46) Point S S = (0.74, 0.46) Point S S = (0.74, 0.46) Point R R = (3.88, 0.42) Point R R = (3.88, 0.42) Point R R = (3.88, 0.42) Point M M = (2.22, 1.14) Point M M = (2.22, 1.14) Point M M = (2.22, 1.14) Point N N = (3.56, 1.12) Point N N = (3.56, 1.12) Point N N = (3.56, 1.12) (ii) text1 = "(ii)"

Fig 2: Parallelogram, Trapezium and Triangles

Figure with different types of parallelogram, trapezium and triangles

Segment f Segment f: Segment [P, Q] Segment g Segment g: Segment [P, S] Segment h Segment h: Segment [S, R] Segment i Segment i: Segment [Q, R] Segment j Segment j: Segment [Q, T] Segment k Segment k: Segment [T, R] Point P P = (2.52, 1.84) Point P P = (2.52, 1.84) Point P P = (2.52, 1.84) Point Q Q = (5.54, 1.84) Point Q Q = (5.54, 1.84) Point Q Q = (5.54, 1.84) Point S S = (1.22, 0.36) Point S S = (1.22, 0.36) Point S S = (1.22, 0.36) Point R R = (4.3, 0.34) Point R R = (4.3, 0.34) Point R R = (4.3, 0.34) Point T Point T: Point on g Point T Point T: Point on g Point T Point T: Point on g (iii) text1 = "(iii)"

Fig 3: Parallelogram, Trapezium and Triangles

Figure with different types of parallelogram, trapezium and triangles

Segment f Segment f: Segment [A, B] Segment g Segment g: Segment [A, D] Segment h Segment h: Segment [D, C] Segment i Segment i: Segment [B, C] Segment j Segment j: Segment [p, R] Segment k Segment k: Segment [R, Q] Point A A = (2.9, 1.14) Point A A = (2.9, 1.14) Point A A = (2.9, 1.14) Point B B = (6.12, 1.14) Point B B = (6.12, 1.14) Point B B = (6.12, 1.14) Point D D = (1.52, -0.48) Point D D = (1.52, -0.48) Point D D = (1.52, -0.48) Point C C = (4.64, -0.48) Point C C = (4.64, -0.48) Point C C = (4.64, -0.48) Point p Point p: Point on i Point p Point p: Point on i Point p Point p: Point on i Point R Point R: Point on g Point R Point R: Point on g Point R Point R: Point on g Point Q Point Q: Point on i Point Q Point Q: Point on i Point Q Point Q: Point on i (iv) text1 = "(iv)"

Fig 4: Parallelogram, Trapezium and Triangles

Figure with different types of parallelogram, trapezium and triangles

Segment f Segment f: Segment [A, D] Segment g Segment g: Segment [A, B] Segment h Segment h: Segment [B, Q] Segment i Segment i: Segment [Q, D] Segment j Segment j: Segment [A, P] Segment k Segment k: Segment [D, C] Point A A = (3.14, 0.7) Point A A = (3.14, 0.7) Point A A = (3.14, 0.7) Point D D = (3.14, -0.76) Point D D = (3.14, -0.76) Point D D = (3.14, -0.76) Point B B = (5.36, 1.36) Point B B = (5.36, 1.36) Point B B = (5.36, 1.36) Point Q Q = (5.36, -1.34) Point Q Q = (5.36, -1.34) Point Q Q = (5.36, -1.34) Point P Point P: Point on h Point P Point P: Point on h Point P Point P: Point on h Point C Point C: Point on h Point C Point C: Point on h Point C Point C: Point on h (v) text1 = "(v)"

Fig 5: Parallelogram, Trapezium and Triangles

Figure with different types of parallelogram, trapezium and triangles

Segment f Segment f: Segment [P, Q] Segment g Segment g: Segment [P, S] Segment h Segment h: Segment [S, R] Segment i Segment i: Segment [Q, R] Segment j Segment j: Segment [A, D] Segment k Segment k: Segment [B, C] Point P P = (3.98, 1.12) Point P P = (3.98, 1.12) Point P P = (3.98, 1.12) Point Q Q = (7.94, 1.14) Point Q Q = (7.94, 1.14) Point Q Q = (7.94, 1.14) Point S S = (3.18, -0.4) Point S S = (3.18, -0.4) Point S S = (3.18, -0.4) Point R R = (7.14, -0.4) Point R R = (7.14, -0.4) Point R R = (7.14, -0.4) Point A Point A: Point on f Point A Point A: Point on f Point A Point A: Point on f Point D Point D: Point on h Point D Point D: Point on h Point D Point D: Point on h Point B Point B: Point on f Point B Point B: Point on f Point B Point B: Point on f Point C Point C: Point on h Point C Point C: Point on h Point C Point C: Point on h (vi) text1 = "(vi)"

Fig 6: Parallelogram, Trapezium and Triangles

Figure with different types of parallelogram, trapezium and triangles

Image title: Figure with different types of parallelogram, trapezium and triangles

Description: Figure with different types of parallelogram, trapezium and triangles

Solution (i):

Segment f Segment f: Segment [A, B] Segment g Segment g: Segment [A, D] Segment h Segment h: Segment [B, C] Segment j Segment j: Segment [D, P] Segment i Segment i: Segment [P, C] Segment P_1 Segment P_1: Segment [D, C] Point A A = (1.24, 1.04) Point A A = (1.24, 1.04) Point A A = (1.24, 1.04) Point B B = (3.86, 1.04) Point B B = (3.86, 1.04) Point B B = (3.86, 1.04) Point D D = (0.74, -0.68) Point D D = (0.74, -0.68) Point D D = (0.74, -0.68) Point C C = (4.2, -0.64) Point C C = (4.2, -0.64) Point C C = (4.2, -0.64) Point P Point P: Point on f Point P Point P: Point on f Point P Point P: Point on f (i) text1 = "(i)"

Trapezium ABCD and Triangle PDC

Give trapezium ABCD and triangle PDC on the same DC and between the same parallel lines AB and DC

It can be observed that trapezium ABCD and triangle PCD have a common base CD and are lying between the same parallel lines AB and CD.

Solution (ii)

Segment f Segment f: Segment [P, Q] Segment g Segment g: Segment [P, S] Segment h Segment h: Segment [Q, R] Segment i Segment i: Segment [S, R] Segment j Segment j: Segment [M, N] Segment k Segment k: Segment [M, S] Segment l Segment l: Segment [N, R] Point P P = (2.06, 1.92) Point P P = (2.06, 1.92) Point P P = (2.06, 1.92) Point Q Q = (5.22, 1.9) Point Q Q = (5.22, 1.9) Point Q Q = (5.22, 1.9) Point S S = (0.74, 0.46) Point S S = (0.74, 0.46) Point S S = (0.74, 0.46) Point R R = (3.88, 0.42) Point R R = (3.88, 0.42) Point R R = (3.88, 0.42) Point M M = (2.22, 1.14) Point M M = (2.22, 1.14) Point M M = (2.22, 1.14) Point N N = (3.56, 1.12) Point N N = (3.56, 1.12) Point N N = (3.56, 1.12) (ii) text1 = "(ii)"

Parallelograms PQRS and Trapezium SMNR

Parallelograms PQRS and trapezium SMNR lie on the same base SR but not between the same parallels

  • As can be seen above, parallelogram PQRS and trapezium SMNR are on same base SR but not between the same parallel lines.

Solution (iii)

Segment f Segment f: Segment [P, Q] Segment g Segment g: Segment [P, S] Segment h Segment h: Segment [S, R] Segment i Segment i: Segment [Q, R] Segment j Segment j: Segment [Q, T] Segment k Segment k: Segment [T, R] Point P P = (2.52, 1.84) Point P P = (2.52, 1.84) Point P P = (2.52, 1.84) Point Q Q = (5.54, 1.84) Point Q Q = (5.54, 1.84) Point Q Q = (5.54, 1.84) Point S S = (1.22, 0.36) Point S S = (1.22, 0.36) Point S S = (1.22, 0.36) Point R R = (4.3, 0.34) Point R R = (4.3, 0.34) Point R R = (4.3, 0.34) Point T Point T: Point on g Point T Point T: Point on g Point T Point T: Point on g (iii) text1 = "(iii)"

Parallelogram ABCD and Triangle RTQ

In this figure, parallelogram ABCD and triangle RTQ lie on the same base and between the same parallel lines QR and PS

As can be seen above, parallelogram PQRS and ΔRTQ lie on the same base QR and between the same parallel lines QR and PS.

Solution (iv)

Segment f Segment f: Segment [A, B] Segment g Segment g: Segment [A, D] Segment h Segment h: Segment [D, C] Segment i Segment i: Segment [B, C] Segment j Segment j: Segment [p, R] Segment k Segment k: Segment [R, Q] Point A A = (2.9, 1.14) Point A A = (2.9, 1.14) Point A A = (2.9, 1.14) Point B B = (6.12, 1.14) Point B B = (6.12, 1.14) Point B B = (6.12, 1.14) Point D D = (1.52, -0.48) Point D D = (1.52, -0.48) Point D D = (1.52, -0.48) Point C C = (4.64, -0.48) Point C C = (4.64, -0.48) Point C C = (4.64, -0.48) Point p Point p: Point on i Point p Point p: Point on i Point p Point p: Point on i Point R Point R: Point on g Point R Point R: Point on g Point R Point R: Point on g Point Q Point Q: Point on i Point Q Point Q: Point on i Point Q Point Q: Point on i (iv) text1 = "(iv)"

Parallelogram ABCD and Triangle PQR

Parallelogram ABCD and triangle PQR do not lie on the same base but between the same parallel lines BC and AD.

In the above figure, parallelogram ABCD and ΔPQR do not lie on the same base but between the same parallel lines BC and AD.

Solution (v)

Segment f Segment f: Segment [A, D] Segment g Segment g: Segment [A, B] Segment h Segment h: Segment [B, Q] Segment i Segment i: Segment [Q, D] Segment j Segment j: Segment [A, P] Segment k Segment k: Segment [D, C] Point A A = (3.14, 0.7) Point A A = (3.14, 0.7) Point A A = (3.14, 0.7) Point D D = (3.14, -0.76) Point D D = (3.14, -0.76) Point D D = (3.14, -0.76) Point B B = (5.36, 1.36) Point B B = (5.36, 1.36) Point B B = (5.36, 1.36) Point Q Q = (5.36, -1.34) Point Q Q = (5.36, -1.34) Point Q Q = (5.36, -1.34) Point P Point P: Point on h Point P Point P: Point on h Point P Point P: Point on h Point C Point C: Point on h Point C Point C: Point on h Point C Point C: Point on h (v) text1 = "(v)"

Quadrilateral ABQD and Trapezium APCD

Quadrilateral ABQD and trapezium APCD lie on the same base AD and between the same parallel lines AD and BQ

Quadrilateral ABQD and trapezium APCD lie on the same base AD and between the same parallel lines AD and BQ.

Solution (vi)

Segment f Segment f: Segment [P, Q] Segment g Segment g: Segment [P, S] Segment h Segment h: Segment [S, R] Segment i Segment i: Segment [Q, R] Segment j Segment j: Segment [A, D] Segment k Segment k: Segment [B, C] Point P P = (3.98, 1.12) Point P P = (3.98, 1.12) Point P P = (3.98, 1.12) Point Q Q = (7.94, 1.14) Point Q Q = (7.94, 1.14) Point Q Q = (7.94, 1.14) Point S S = (3.18, -0.4) Point S S = (3.18, -0.4) Point S S = (3.18, -0.4) Point R R = (7.14, -0.4) Point R R = (7.14, -0.4) Point R R = (7.14, -0.4) Point A Point A: Point on f Point A Point A: Point on f Point A Point A: Point on f Point D Point D: Point on h Point D Point D: Point on h Point D Point D: Point on h Point B Point B: Point on f Point B Point B: Point on f Point B Point B: Point on f Point C Point C: Point on h Point C Point C: Point on h Point C Point C: Point on h (vi) text1 = "(vi)"

Parallelogram PQRS and Parallelogram ABCD

Parallelogram PQRS and parallelogram ABCD do not lie on the same base SR but between the same parallel lines SR and PQ.

Parallelogram PQRS and parallelogram ABCD do not lie on the same base SR but between the same parallel lines SR and PQ.

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