Q1
(a) In the figure (1) given below, D, E and F are mid-points of the sides BC, CA and AB respectively of ∆ ABC. If AB = 6 cm, BC = 4.8 cm and CA= 5.6 cm, find the perimeter of (i) the trapezium FBCE (ii) the triangle DEF.
(b) In the figure (2) given below, D and E are mid-points of the sides AB and AC respectively. If BC = 5.6 cm and∠B = 72°, compute (i) DE (ii)∠ADE.
(c) In the figure (3) given below, D and E are mid-points of AB, BC respectively and DF || BC. Prove that DBEF is a parallelogram. Calculate AC if AF = 2.6 cm.
solutions
(b) In the figure (2) given below, D and E are mid-points of the sides AB and AC respectively. If BC = 5.6 cm and∠B = 72°, compute (i) DE (ii)∠ADE.
(c) In the figure (3) given below, D and E are mid-points of AB, BC respectively and DF || BC. Prove that DBEF is a parallelogram. Calculate AC if AF = 2.6 cm.
solutions
Q2
Prove that the four triangles formed by joining in pairs the mid-points of the sides c of a triangle are congruent to each other.
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solutions
Q3
If D, E and F are mid-points of sides AB, BC and CA respectively of an isosceles triangle ABC, prove that ∆DEF is also F, isosceles
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solutions
Q4
The diagonals AC and BD of a parallelogram ABCD intersect at O. If P is the mid-point of AD, prove that
(i) PQ || AB
(ii) PO=1/2CD.
solutions
(i) PQ || AB
(ii) PO=1/2CD.
solutions
Q5
In the adjoining figure, ABCD is a quadrilateral in which P, Q, R and S are mid-points of AB, BC, CD and DA respectively. AC is its diagonal. Show that
(i) SR || AC and SR =1/2AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
solutions
(i) SR || AC and SR =1/2AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.
solutions
Q6
Show that the quadrilateral formed by joining the mid-points of the adjacent sides of a square, is also a square,
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solutions
Q7
In the adjoining figure, AD and BE are medians of ∆ABC. If DF U BE, prove that CF = 1/4 AC.
solutions
solutions
Q8
(a) In the figure (1) given below, ABCD is a parallelogram. E and F are mid-points of the sides AB and CO respectively. The straight lines AF and BF meet the straight lines ED and EC in points G and H respectively. Prove that
(i) ∆HEB = ∆HCF
(ii) GEHF is a parallelogram.
(b) In the diagram (2) given below, ABCD is a parallelogram. E is mid-point of CD and P is a point on AC such that PC = 1/4 AC. EP produced meets BC at F. Prove that
(i) F is mid-point of BC (ii) 2EF = BD
solutions
(i) ∆HEB = ∆HCF
(ii) GEHF is a parallelogram.
(b) In the diagram (2) given below, ABCD is a parallelogram. E is mid-point of CD and P is a point on AC such that PC = 1/4 AC. EP produced meets BC at F. Prove that
(i) F is mid-point of BC (ii) 2EF = BD
solutions
Q9
ABC is an isosceles triangle with AB = AC. D, E and F are mid-points of the sides BC, AB and AC respectively. Prove that the line segment AD is perpendicular to EF and is bisected by it.
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solutions
Q10
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solutions
Q11
(a) In the quadrilateral (1) given below, AD = BC, P, Q, R and S are mid-points of AB, BD, CD and AC respectively. Prove that PQRS is a rhombus.
(b) In the figure (2) given below, ABCD is a kite in which BC = CD, AB = AD, E, F, G are mid-points of CD, BC and AB respectively. Prove that:
(i) ∠EFG = 90 (ii) The line drawn through G and parallel to FE bisects DA
solutions
(b) In the figure (2) given below, ABCD is a kite in which BC = CD, AB = AD, E, F, G are mid-points of CD, BC and AB respectively. Prove that:
(i) ∠EFG = 90 (ii) The line drawn through G and parallel to FE bisects DA
solutions
Q12
In the adjoining figure, the lines l, m and n are parallel to each other, and G is mid-point of CD. Calculate:
(i) BG if AD = 6 cm
(ii) CF if GE = 2.3 cm
(iii) AB if BC = 2.4 cm
(iv) ED if FD = 4.4 cm.
solutions
(i) BG if AD = 6 cm
(ii) CF if GE = 2.3 cm
(iii) AB if BC = 2.4 cm
(iv) ED if FD = 4.4 cm.
solutions