(English Medium) ICSE Class 9 - CISCE Question Bank Solutions

In a quadrilateral ABCD, ∠B = 90° and ∠D = 90°.
Prove that: 2AC² - AB² = BC² + CD² + DA²

In a rectangle ABCD,
prove that: AC² + BD² = AB² + BC² + CD² + DA².

In triangle ABC, ∠B = 90^o and D is the mid-point of BC.

Prove that: AC² = AD² + 3CD².

M andN are the mid-points of the sides QR and PQ respectively of a PQR, right-angled at Q.
Prove that:
(i) PM² + RN² = 5 MN²
(ii) 4 PM² = 4 PQ² + QR²
(iii) 4 RN² = PQ² + 4 QR²(iv) 4 (PM² + RN²) = 5 PR²

Construct a parallelogram ABCD, when:
AB = 4.4 cm, AD = 6.2 cm and AC = 4.8 cm.

Draw parallelogram ABCD with the following data:
AB = 6 cm, AD = 5 cm and ∠DAB = 45^o.
Let AC and DB meet in O and let E be the mid-point of BC. Join OE.
Prove that:
(i) OE // AB
(ii) OE = `1/2` AB.

Construct a parallelogram ABCD, when:
Diagonal AC = 6.4 cm, diagonals BD = 8.2 cm and angle between the diagonals = 60°.

The perpendicular distance between the pair of opposite sides of a parallelogram are 3 cm and 4 cm, and one of its angles measures 60^o. Using a ruler and compasses only,
construct the parallelogram.

Construct a parallelogram ABCD, when:
AB = 5.8 cm, diagonal AC = 8.2 cm and diagonal BD = 6.2 cm.

Construct a parallelogram ABCD, when:
AB = 6.0 cm, AD = 5.0 cm and ∠A = 45°.

Using ruler and compasses only, construct a parallelogram ABCD using the following data: AB = 6 cm, AD = 3 cm and ∠DAB = 45^o. If the bisector of ∠DAB meets DC at P,
prove that ∠APB is a right angle.

Construct a parallelogram ABCD, when:
Base AB = 6.5 cm, BC = 4 cm and the altitude corresponding to AB = 3.1 cm.

Construct a parallelogram ABCD, when:

AB = 4.5 cm, ∠B = 120° and the distance between AB and DC = 3.0 cm.

Construct a parallelogram ABCD, when:
Base BC = 5.6 cm, diagonal BD = 6.5 cm and altitude = 3.2 cm.

In the given figure, if the area of triangle ADE is 60 cm², state, given reason, the area of :
(i) Parallelogram ABED;
(ii) Rectangle ABCF;
(iii) Triangle ABE.

The given figure shows the parallelograms ABCD and APQR.
Show that these parallelograms are equal in the area.
[ Join B and R ]

The given figure shows a rectangle ABDC and a parallelogram ABEF; drawn on opposite sides of AB.
Prove that: 
(i) Quadrilateral CDEF is a parallelogram;
(ii) Area of the quad. CDEF
= Area of rect. ABDC + Area of // gm. ABEF.

In the given figure, ABCD is a parallelogram; BC is produced to point X.
Prove that: area ( Δ ABX ) = area (`square`ACXD )

In the given figure, AD // BE // CF.
Prove that area (ΔAEC) = area (ΔDBF)

ABCD is a trapezium with AB // DC. A line parallel to AC intersects AB at point M and BC at point N.
Prove that: area of Δ ADM = area of Δ ACN.