English Medium कक्षा ९ - CBSE Question Bank Solutions for Mathematics

Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.

If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD.

A chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in major segment.

In the following figure, ∠ACB = 40º. Find ∠OAB.

In the following figure, AB and CD are two chords of a circle intersecting each other at point E. Prove that ∠AEC = `1/2` (Angle subtended by arc CXA at centre + angle subtended by arc DYB at the centre).

A circle has radius `sqrt(2)` cm. It is divided into two segments by a chord of length 2 cm. Prove that the angle subtended by the chord at a point in major segment is 45°.

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:

1. OB = OC
2. AO bisects ∠A

In ΔABC, AD is the perpendicular bisector of BC (see the given figure). Show that ΔABC is an isosceles triangle in which AB = AC.

ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see the given figure). Show that these altitudes are equal.

ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that

1. ΔABE ≅ ΔACF
2. AB = AC, i.e., ABC is an isosceles triangle.

ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.

ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see the given figure). Show that ∠BCD is a right angle.

Show that the angles of an equilateral triangle are 60° each.

A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

In the given figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.

ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD.

If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.

If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see the given figure). Prove that ∠ACP = ∠QCD.

If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lie on the third side.