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

In an isosceles triangle ABC; AB = AC and D is the point on BC produced.

Prove that: AD² = AC² + BD.CD.

In figure AB = BC and AD is perpendicular to CD.
Prove that: AC² = 2BC. DC.

ABC is a triangle, right-angled at B. M is a point on BC.

Prove that: AM² + BC² = AC² + BM²

Diagonals of rhombus ABCD intersect each other at point O.

Prove that: OA² + OC² = 2AD² - `"BD"^2/2`

In the following figure, OP, OQ, and OR are drawn perpendiculars to the sides BC, CA and AB respectively of triangle ABC.

Prove that: AR² + BP² + CQ² = AQ² + CP² + BR²

O is any point inside a rectangle ABCD.
Prove that: OB² + OD² = OC² + OA².

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 quadrilateral ABCD, when:
AB = 3.2 cm, BC = 5.2 cm, CD = 6.2 cm, DA = 4.2 cm and BD = 5.2 cm.

Construct a quadrilateral ABCD, when:
AB = 7.2 cm, BC = 5.8 cm, CD = 6.3 cm, AD = 4.3 cm and angle A = 75°.

Construct a quadrilateral ABCD, when:
∠ A = 90^o, AB = 4.6 cm, BD = 6.4 cm, AC = 6.0 cm and CD = 4.2 cm.

Construct a quadrilateral ABCD, when:
AB = 3.8 cm, AC = 4.8 cm, AD = 2.8 cm, angle A = 105° and angle B = 60°.

Construct a quadrilateral ABCD, when:
BC = 7.5 cm AC = 5.8 cm, AD = 3.6 cm, CD = 4.2 cm and angle A = 120°.

Construct a quadrilateral ABCD, when:
AD = AB = 4 cm, BC = 2.8 cm, CD = 2.5 cm and angle BAD = 45°.

Construct a quadrilateral ABCD, when:
AB = 6.3 cm, BC = CD = 4.2 cm and ∠ABC = ∠BCD = 90°.

Using ruler and compasses only, construct the quadrilateral ABCD, having given AB = 5 cm, BC = 2.5 cm CD = 6 cm, ∠BAD = 90^o and diagonal BD = 5.5 cm.

Using ruler and compasses only, construct the quadrilateral ABCD, having given AB = 5 cm, BC = 2.5 cm, CD = 6 cm. angle BAD = 90^o and the diagonal AC = 5.5 cm.

The perimeter of a triangle ABC is 37 cm and the ratio between the lengths of its altitudes be 6: 5: 4. Find the lengths of its sides.
Let the sides be x cm, y cm, and (37 - x - y) cm. Also, let the lengths of altitudes be 6a cm, 5a cm, and 4a cm.