Question

4) In a parallelogram ABCD, AB = 10 cm and AD = 6 cm. The bisector of ZA meets DC in E. AE and BC produced meet at F. Find the length of CF. 5) P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Prove that PQRS is a rhombus. 6) P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD such that AC 1 BD. Prove that PQRS is a rectangle. 7) P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC = BD and AC I BD. Prove that PQRS is a square. 8) P and Q are the mid-points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram. 9) ABCD is a quadrilateral in which AB || DC and AD = BC. Prove that ZA = 2B and C = 2D. 10) The angle between two altitudes of a parallelogram through the vertex of an obtuse angle of the parallelogram is 60°. Find the angles of the parallelogram. 11) ABCD is a trapezium in which AB || DC and ZA = 2B = 45°. Find angles C and D of the trapezium 12) E is the mid-point of a median AD of A ABC and BE is produced to meet AC at F. Show that AF = 1/3 AC. 13) Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle. 14) E and F are respectively the mid-points of the non-parallel sides AD and BC of a trapezium ABCD. Prove that EF || AB and EF = /2 (AB + CD). 15) D, E and F are respectively the mid-points of the sides AB, BC and CA of a AABC. Prove that by joining these mid-points D, E and F, the AABC is divided into four congruent triangles.​

Answer 1

Step-by-step explanation:

According to the question,

We have,

ABCD is a parallelogram

AB = 10 cm

AD = 6cm.

The bisector of ∠A meets DC at E.

AE and BC produced meet at F.

Since, AF bisects ∠A,

We get,

∠BAE = ∠EAD … (1)

∠EAD = ∠EFB … (2) [Alternate angles]

From equations (1) and (2),

We get,

∠BAE = ∠EFB

Since sides opposite to equal angles are equal,

We get,

BF = AB

Here, AB = 10 cm

So, BF = 10 cm

⇒ BC + CF = 10 cm

6 cm + CF = 10 cm [BC = AD = 6 cm, opposite sides of a parallelogram]

⇒ CF = 10 – 6 cm = 4 cm

⇒ CF = 4 cm