Class 9 Quadrilaterals Important Questions

Important Questions

Quadrilaterals Important Questions

1. Show that each angle of a rectangle is a right angle.
2. Find all the angles of a parallelogram if one angle is 60°.
3. Two parallel lines l and m are intersected by a transversal p (See in fig.). Show that the quadrilateral formed by the bisectors of interior angles is a rectangle.
   
4. The angles of quadrilateral are in the ratio 2 : 4 : 8 : 12. Find all the angles of the quadrilateral.
5. The diagonals of which quadrilateral are equal and bisect each other at 90°?
6. If the diagonals of a parallelogram are equal, then show that it is a rectangle.
7. In a trapezium ABCD, AB∥CD. Calculate ∠C and ∠D if ∠A = 55° and ∠B = 70°.
8. L, M, N, K are mid-points of sides BC, CD, DA and AB respectively of square ABCD, prove that DL, DK, BM and BN enclose a rhombus.
9. In a parallelogram ABCD find the measure of all the angles if one its angles is 15° less than twice the smallest angle.
10. If ABCD is a parallelogram, then prove that ar(∆ABD) = ar(∆BCD) = ar(∆ABC) = ar(∆ACD) = ½ar(||^gm ABCD)
11. If AD is a median of a triangle ABC, then prove that triangles ADB and ADC are equal in area. If G is the mid point of median AD, prove that ar(∆BGC) = 2ar(∆AGC).
12. ABCD is a trapezium in which AB || CD and AD = BC (see Fig.). Show that
    1. ∠ A = ∠ B
    2. ∠ C = ∠ D
    3. ∆ ABC ≅ ∆ BAD
    4. diagonal AC = diagonal BD
       [Hint : Extend AB and draw a line through C parallel to DA intersecting AB produced at E.]
       
13. Diagonals AC and BD of a quadrilateral ABCD intersect each other at P. Show that: ar(∆APB) x ar(∆CPD) = ar(∆APD) x ar(∆BPC).
14. In a ∆ABC, P and Q are respectively, the mid-points of AB and BC and R is the mid-point of AP. Prove that
    1. ar(∆PBQ) = ar(∆ARC)
    2. ar(∆PRQ) = ½ ar(∆ARC)
    3. ar(∆RQC) = ⅜ ar(∆ABC).
15. D is the mid-point of side BC of ∆ABC and E is the mid-point of BD. If O is the mid-point of AE, prove that ar(∆BOE) = ⅛ ar(∆ABC).
16. In ∆ ABC, D, E and F are respectively the mid-points of sides AB, BC and CA (see Fig.). Show that ∆ ABC is divided into four congruent triangles by joining D, E and F.
    
17. ABC is a triangle right angled at C. A line through the mid-point M of hypotenuse AB and parallel to BC intersects AC at D. Show that
    1. D is the mid-point of AC
    2. MD ⊥ AC
    3. CM = MA =½AB.
18. In the figure, ABC is a right triangle right angled at A, BCED, ACFG and ABMN are squares on the sides BC, CA and AB respectively. Line segment AX ⊥ DE meets BC at Y. Show that
    
    1. ∆MBC ≅ ∆ABD
    2. ar(BYXD) = ar(∆MBC)
    3. ar(BYXD) = ar(ABMN)
    4. ∆FCB ≅ ∆ACE
    5. ar(CYXE) = 2ar(∆FCB)
    6. ar(CYXE) = ar(∆CFG)
    7. ar(BCED) = ar(AMBN) + ar(ACFG)
19. ABCD is a quadrilaeral in which the bisectors of ∠A and ∠C meet DC produced at Y and BA produced at X respectively. Prove that ∠X + ∠Y = ½(∠A + ∠C).
    
20. ABCD is a parallelogram. P is the mid-point of AB. BD and CP intersect at Q such that CQ : QP = 3 : 1. If ar(∆PBQ) = 10 cm², find the area of parallelogram ABCD.
21. A triangle and a parallelogram are on the same base and between the same parallels. The ratio of the areas of triangle and parallelogram is
    1. 1 : 1
    2. 1 : 2
    3. 2 : 1
    4. 1 : 3