In triangle ABC, the medians AD and BE are perpendicular to each other. If AD = 9 and BE = 12, what is the area of triangle ABC?

A chord of length 16 is drawn in a circle, at a perpendicular distance of 6 from the centre. A second chord, of length 12, is drawn parallel to the first one, on the opposite side of the centre. What is the distance (in the same units) between the two chords? Enter the answer as an integer. - -

In a trapezium ABCD, AB ∥DC, AD ⊥AB, and AB = 2DC. A circle is inscribed in the trapezium and touches all four sides. If the radius of the circle is 4 cm, then the area of the trapezium, in sq. cm, is

In a trapezium ABCD, AB ∥CD, AD ⊥AB, and AB = 2CD. A circle touches all four sides of the trapezium, and its radius is 5 cm. What is the area, in sq. cm, of the trapezium?

A circle has centre O and radius 5 cm. From an external point P, where OP = 13 cm, two tangents PA and PB are drawn to the circle, touching it at A and B, respectively. Find the area, in sq. cm, of quadrilateral OAPB. 3.

ABCD is a trapezium with AB ∥CD, AB = 8 and CD = 18. The diagonals AC and BD intersect at P. If the area of triangle APB is 32 square units, then the area of the trapezium ABCD (in square units) is

In the coordinate plane, A = (0, 0), B = (12, 0), and C = (0, 9). A line passing through (4, 3) cuts AB at P and AC at Q, with P and Q lying on the respective segments. What is the minimum possible area of triangle APQ?

In triangle ABC, AB = AC = 25 cm and BC = 30 cm. A circle has its centre on the altitude from A to BC, touches both AB and AC, and passes through the midpoint of BC. What is the radius of the circle, in cm?

A line cuts the positive coordinate axes at (a, 0) and (0, b), where a and b are positive integers. The line passes through (4, 3). Find the sum of the areas of all possible triangles formed by such a line and the coordinate axes.

A right circular cone of base radius 6 cm and height 24 cm is melted (with no loss of material) and recast into a right circular cylinder of base radius 4 cm. A hemispherical cavity of radius 4 cm, fully contained inside the cylinder, is then carved out of one of the two flat circular ends of the cylinder. The total surface area of the resulting solid can be written as k π square centimetres, where k is a positive integer. Find k.

The points A(-4, 1), B(2, -3), and C(9, 2) are three consecutive vertices of a parallelogram ABCD. If the diagonal BD meets the x - axis at (t, 0), then the value of t is

In a circle of radius 10 cm, two parallel chords PQ and RS lie on opposite sides of the centre. The distance of PQ from the centre and the distance of RS from the centre are in the ratio 3: 4. If the length of PQ is 16 cm, find the area, in sq. cm, of quadrilateral PQRS.

In triangle ABC, AB = 13, BC = 14 and CA = 15. Let D be the point on BC where the internal bisector of angle A meets BC, and let M be the midpoint of segment AD. The ratio of the area of triangle BMC to the area of triangle ABC is:

A right circular cone has base radius 6 cm and height 8 cm. A sphere is inscribed inside the cone such that it touches the base of the cone and is also tangent to the slant surface all around. The volume of this sphere (in cm3) is:

In right triangle ABC, ∠A = 90◦, AB = 24 cm and AC = 32 cm. A square is placed inside the triangle such that one side of the square lies on hypotenuse BC, and the two vertices of the opposite side lie on AB and AC, respectively. Find the side length of the square, in cm.