CBSE class 9 Areas of Parallelograms and Triangles Exercise 9 3

http://www.learncbse.in/ncert-solutions-for-class-9-maths/

00:02 CBSE class 9 Areas of Parallelograms and Triangles Q1. In Fig.9.23, E is any point o.n median AD of a Triangle ABC. Show that ar (ABE) = ar (ACE).

00:30 Defination of Median : A median of a triangle is a line segment that joins the vertex of a triangle to the midpoint of the opposite side

01:06 Apply Area of triangle formula:Area of a triangle is the product of its base and the corresponding altitude.

03:18 Apply Area Axiom:If a planar region formed by a figure T is made up of two non-overlapping planar regions formed by figures P and Q, then ar(T)= ar (P)+ ar(Q)

05:25 CBSE class 9 Areas of Parallelograms and Triangles Q2. In a triangle ABC, E is the mid-point of median AD. Show that ar (BED) = 1/4 ar(ABC)

06:23 Median:A median of a triangle is a line segment that joins the vertex of a triangle to the midpoint of the opposite side

06:38 Apply Area Theorem:A median of a triangle divides it into two triangles of equal areas.

08:22 CBSE class 9 Areas of Parallelograms and Triangles Q3. Show that the diagonals of a parallelogram divide it into four triangles of equal area.

09:55 Median of Triangle

09:59 A median of a triangle divides it into two triangles of equal areas.

12:40 Q4. In Fig. 9.24, ABC and ABD are two triangles on the same base AB. If line- segment CD is bisected by AB at O, show that ar(ABC) = ar (ABD).

13:48 A median of a triangle divides it into two triangles of equal areas.

15:42 Q5. D, E and F are respectively the mid-points of the sides BC, CA and AB of a Triangle ABC. Show that (i) BDEF is a parallelogram. (ii) ar (DEF) = 1/4 ar (ABC) (iii) ar (BDEF) = 1/2 ar (ABC).

17:51 Apply Mid point Theorem:The line-segment joining the mid-points of any two sides of a triangle is parallel to the third side and is half of it.

19:43 Use Properties of Parallelogram:In a parallelogram, pair of opposite sides are equal & Opposite angels are equal.

20:11 (ii)

26:09 Adding two results to get the desired results

29:26 Apply properties of paralleogram:Diagonals of a parallelogram bisect each other at right angles

24:13 Q6. In Fig. 9.25, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that: (i) ar (DOC) = ar (AOB) (ii) ar (DCB) = ar (ACB) (iii) DA || CB or ABCD is a parallelogram. [ Hint : From D and B, draw perpendiculars to AC.]

27:00 Apply area of triangle

31:39 CBSE class 9 Areas of Parallelograms and Triangles Q7. D and E are points on sides AB and AC respectively of triangle ABC such that ar (DBC) = ar (EBC). Prove that DE || BC.

33:15 Find Area of Triangle:Area of a triangle is half the product of its base (or any side ) and the corresponding altitude (or height

34:04 CBSE class 9 Areas of Parallelograms and Triangles Q8. XY is a line parallel to side BC of a triangle ABC. If BE || AC and CF || AB meet XY at E and F respectively, show that ar (ABE) = ar (ACF)

36:01 Apply Propeerties of Parallelogram

41:52 Q9. The side AB of a parallelogram ABCD is produced to any point P. A line through A and parallel to CP meets CB produced at Q and then parallelogram PBQR is completed (see Fig. 9.26). Show that ar (ABCD) = ar (PBQR).[ Hint : Join AC and PQ. Now compare ar (ACQ) and ar (APQ).]

42:37 Apply Prallelogram Theorem:If a parallelogram and a triangle are on the same base and between the same parallels, then area of the triangle is half the area of the parallelogram.

45:17 Q10. Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O.Prove that ar (AOD) = ar (BOC).

47:02 Apply Prallelogram Theorem:Triangles on the same base (or equal bases) and between the same parallels are equal in area

47:55 Q11. In Fig. 9.27, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that (i) ar (ACB) = ar (ACF) (ii) ar (AEDF) = ar (ABCDE)

49:23 Use Prallelogram Theorem:Triangles on the same base (or equal bases) and between the same parallels are equal in area

51:12 Q13. ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar (ADX) = ar (ACY). [ Hint : Join CX.]

51:49 Procedure for construction to solve the problem.

54:30 Triangle Theorem:Two triangles on the same base(or equal bases) and between the same parallels are equal in area .

55:08 CBSE class 9 Areas of Parallelograms and Triangles Q14. In Fig.9.28, AP || BQ || CR. Prove that ar (AQC) = ar (PBR).

56:05 Triangle Theorem

57:55 Q15. Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar (AOD) = ar (BOC). Prove that ABCD is a trapezium.

59:34 Use Triangle Theorem:Two triangles on the same base(or equal bases) and between the same parallels are equal in area

01:00:20 CBSE class 9 Areas of Parallelograms and Triangles Q16. In Fig.9.29, ar (DRC) = ar (DPC) and ar (BDP) = ar (ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums.

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