English Medium Class 9 - CBSE Question Bank Solutions for Mathematics

ABC and ADC are two right triangles with common hypotenuse AC. Prove that ∠CAD = ∠CBD.

Prove that a cyclic parallelogram is a rectangle.

Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Let the vertex of an angle ABC be located outside a circle and let the sides of the angle intersect equal chords AD and CE with the circle. Prove that ∠ABC is equal to half the difference of the angles subtended by the chords AC and DE at the centre.

Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals.

ABCD is a parallelogram. The circle through A, B and C intersect CD (produced if necessary) at E. Prove that AE = AD.

AC and BD are chords of a circle which bisect each other. Prove that (i) AC and BD are diameters; (ii) ABCD is a rectangle.

Two congruent circles intersect each other at points A and B. Through A any line segment PAQ is drawn so that P, Q lie on the two circles. Prove that BP = BQ.

In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC.

Two chords AB and CD of lengths 5 cm 11cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.

The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre?

Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90°-A, 90° − `1/2 A, 90° − 1/2 B, 90° − 1/2` C.

In a ΔABC, if ∠A=l20° and AB = AC. Find ∠B and ∠C. 

In Figure AB = AC and ∠ACD =105°, find ∠BAC.  

Find the measure of each exterior angle of an equilateral triangle. 

If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other. 

In figure, AB = AC and DB = DC, find the ratio ∠ABD : ∠ACD 

Determine the measure of each of the equal angles of a right-angled isosceles triangle.

AB is a line seg P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. 10.26). Show that the line PQ is perpendicular bisector of AB. 

In Fig. 10.40, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT