Nootan solutions for माठेमटिक्स [इंग्रजी] इयत्ता ९ आयसीएसई chapter 9 - Mid-point Theorem [Latest edition]

ΔABC is an isosceles triangle in which AB = AC. If P, Q and R are the mid-points of the sides AB, BC and CA, respectively, prove that ΔPQR is also an isosceles triangle.

In the adjoining figure; P, Q and R are the mid-points of the sides BC, CA and AB, respectively, of ΔABC. Prove that ◻RQPB is a parallelogram.

◻PQRS is a rectangle. If A, B and C are the mid-points of sides PQ, PS and QR, respectively. Prove that AB + AC = `1/2` (PR + SQ).

Show that the straight lines joining the mid-points of the opposite sides of a quadrilateral bisect each other.

In ΔАВС, M and N are the mid-points of AB and AC respectively and R be any point on BC. Use intercept theorem, prove that MN bisects AR.

In the adjoining figure, ◻ABCD is a parallelogram. P is the mid-point of CD and DQ || PB meets CB produced at R.

Prove that:

1. AD = `1/2` CR
2. DR = 2 PB

In a trapezium ABCD, AB || CD. M and N are two points on AD dividing it into three equal parts. Line segments MP and NQ are parallel to AB which meet BC at P and Q, respectively. Prove that BP = PQ = CQ, i.e., P and Q divide BC into three equal parts.

In the adjoining figure, ABCD is a trapezium in which AB || DC. If M and N are the midpoints of BD and AC, respectively, prove that MN = `1/2` (CD – AB).

In the adjoining figure, ABCD is a parallelogram in which P is the mid-point of DC. If Q is any point on AC such that CQ = `1/4` CA and PQ produced meet BC at R, prove that R is the mid-point of BC.

Show that the quadrilateral formed by joining the mid-points of the adjacent sides of a square is also a square.

In a parallelogram ABCD, the diagonals AC and BD intersect at point O. If P is the mid-point of BC, prove that

1. PO || AB
2. PO = `1/2` AB

In the adjoining figure; D, E and F are the mid-points of BC, CA and AB, respectively. If AB = 6 cm, BC = 8 cm and AC = 5.6 cm, find :

1. perimeter of ΔDEF
2. perimeter of ◻ABDE

ABCD is a rhombus. EABF is a straight line such that EA = AB = BF. Prove that ED and FC when produced, meet at right angles.

In the adjoining figure; AP = `1/2` AB and D is the mid-point of AB, Q is the mid-point of PR and DR || BS.

Prove that:

1. AQ || DR
2. PQ = QR = RS

In the adjoining figure; `CE = 1/2 AC` and D is the mid-point of BC.

Prove that:

1. DE = 2DF
2. AB = 4CQ

In ΔAВC; D and E are two points on AB such that AD = DE = EB. Through D and E, lines are drawn parallel to AC meet BC at N and M, respectively. Through N and M, lines are drawn parallel to AB meet AC at Q and P. Prove that AP = PQ = CQ.

In the following figure, l || m || n and AB = BC, find:

1. PQ if QR = 4 cm
2. BO if CR = 6 cm

In the trapezium ABCD, P and Q are the mid-points of non-parallel sides AD and BC, respectively. Then PQ is equal to ______.

In the trapezium ABCD, P and Q are the mid-points of AC and BD respectively AB || CD with AB > CD. Then PQ is equal to ______.

In a parallelogram ABCD, the diagonals AC and BD intersect at point M. If P is the mid-point of side AD, then PM is equal to ______.

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a rhombus, if ______.

The quadrilateral formed by joining the mid-point of a quadrilateral ABCD, taken in order is a rectangle, if ______.

In ΔАВC, the mid-points of the sides BC, CA and AB are P, Q and R respectively. PR and BQ meet at X. CR and PQ meet at Y. Then XY is equal to ______.

ΔABC is an isosceles triangle. P, Q and R are the mid-points of the sides BC, CA and AB, respectively. Then ΔPQR is ______.