NCERT Class 9 Solutions: Quadrilaterals (Chapter 8) Exercise 8.2 – Part 2

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  • Mid-point

The triangle of ADE and EFC, in this triangle the mid point …

The Triangle of ADE and EFC

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In a triangle, the line segment that joins the midpoints of the two sides of a triangle is parallel to the third side and is half of it

Q-3 ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

Solution:

ABCD is a Rectangle Also PQRS is a Rhombus Quadrilateral

  • Give, ABCD is a ABCD is a rectangle and PQRS is a rhombus quadrilateral .P, Q, R

  • And S is mid-points of the sides AB, BC, CD and DA respectively.

In

  • P and Q are the mid-points of AB and BC respectively

  • Thus, and (Mid-point theorem) …….equation (1)

In,

  • S, R are the mid-points of AD and DC respectively

  • and (Mid-point theorem)……….equation (2)

From equation (1) and (2)

  • So and

As in quadrilateral PQRS one pair of opposite sides is equal and parallel to each other, so, it is a parallelogram.

  • and (Opposite sides of parallelogram) ……..equation (3)

Now, In,

  • Q and R are mid points of side BC and CD respectively.

  • Thus, and (Mid-point theorem) ……..equation (4)

  • (Diagonals of a rectangle are equal)………..equation (5)

From equations (1), (2), (3), (4) and (5),

  • So, PQRS is a rhombus.

Q-4 ABCD is a trapezium in which, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see Fig.). Show that F is the mid-point of BC.

ABCD is a Trapezium

Solution:

  • Give, ABCD is a trapezium in which,

  • BD is a diagonal and E is the mid-point of AD.

  • A line is drawn through E parallel to AB intersecting BC at F

Proof

  • F is the mid-point of BC.

  • BD intersected EF at G.

  • In ,

  • E is the mid-point of AD and also

  • Thus, G is the mid-point of BD (Converse of midpoint theorem)

Now, In ΔBDC,

  • G is the mid-point of BD and also GF || AB || DC.

  • Thus, F is the mid-point of BC (Converse of midpoint theorem)