# Geometry

### In given figure ∠1 = ∠2 and ΔNSQ ≅ ΔMTR , then prove that ΔPTS ~ ΔPRQ

In given figure ∠1 = ∠2 and ΔNSQ ≅ ΔMTR , then prove that ΔPTS ~ ΔPRQ. ### Find a relation between x and y if the points A(x, y), B(–4, 6) and C(–2, 3)

Find a relation between x and y if the points A(x, y), B(–4, 6) and C(–2, 3) are collinear.

### Area of a triangle whose vertices are (1, –1) (–4, 6) and (–3, –5)

Find the area of a triangle whose vertices are given as (1, –1) (–4, 6) and (–3, –5).

### Two right triangles ABC and DBC are drawn on the same hypotenuse

Two right triangles ABC and DBC are drawn on the same hypotenuse BC and on the same side of BC. If AC and BD intersect at P, prove that AP × PC = BP × DP.

### Diagonals of a trapezium PQRS intersect each other at the point O

Diagonals of a trapezium PQRS intersect each other at the point O, PQ ∥ RS and PQ = 3RS. Find the ratio of the areas of triangles POQ and ROS.

### In Figure, PQ and RS are two parallel tangents to a circle with centre O

In Figure, PQ and RS are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting PQ at A and RS at B. Prove that ∠AOB = 90º. ### Find the ratio in which the line x – 3y = 0 divides the line segment

Find the ratio in which the line x – 3y = 0 divides the line segment joining the points (–2, –5) and (6, 3). Find the coordinates of the point of intersection.

### X is a point on the side BC of ΔABC. XM and XN are drawn parallel

X is a point on the side BC of ΔABC. XM and XN are drawn parallel to AB and AC respectively meeting AB in N and AC in M. MN produced meets CB produced at T. Prove that TX2 = TB × TC

### In Figure, ABC is a triangle in which ∠B = 90º, BC = 48 cm and AB = 14 cm

In Figure, ABC is a triangle in which ∠B = 90º, BC = 48 cm and AB = 14 cm. A circle is inscribed in the triangle, whose centre is O. Find radius r of in-circle. ### Find the linear relation between x and y such that P(x, y) is equidistant

Find the linear relation between x and y such that P(x, y) is equidistant from the points A(1, 4) and B(–1, 2).

### If coordinates of two adjacent vertices of a parallelogram are (3, 2), (1, 0)

If coordinates of two adjacent vertices of a parallelogram are (3, 2), (1, 0) and diagonals bisect each other at (2, –5), find coordinates of the other two vertices.

### If the area of triangle with vertices (x, 3), (4, 4) and (3, 5) is 4 square units

If the area of triangle with vertices (x, 3), (4, 4) and (3, 5) is 4 square units, find x.

### In figure, AB is a chord of length 8 cm of a circle of radius 5 cm

In figure, AB is a chord of length 8 cm of a circle of radius 5 cm. The tangents to the circle at A and B intersect at P. Find the length of AP. ### Prove that the ratio of the areas of two similar triangles is equal to

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

### Find the ratio in which P(4, m) divides the line segment

Find the ratio in which P(4, m) divides the line segment joining the points A(2, 3) and B(6, –3). Hence find m.

### If A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral

If A(–5, 7), B(–4, –5), C(–1, –6) and D(4, 5) are the vertices of a quadrilateral, find the area of the quadrilateral ABCD.

### If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD

If A(–2, 1), B(a, 0), C(4, b) and D(1, 2) are the vertices of a parallelogram ABCD, find the values of a and b. Hence find the lengths of its sides.

### Prove that the area of an equilateral triangle described on one side of the square

Prove that the area of an equilateral triangle described on one side of the square is equal to half the area of the equilateral triangle described on one of its diagonal.

### If the area of two similar triangles are equal, prove that they are congruent

If the area of two similar triangles are equal, prove that they are congruent.

### Find the area of the shaded region, where arcs drawn with centres A, B, C and D intersect

Find the area of the shaded region in Figure, where arcs drawn with centres A, B, C and D intersect in pairs at mid-points P, Q, R and S of the sides AB, BC, CD and DA respectively of a square ABCD of side 12 cm. [Use π = 3.14] ### In an equilateral triangle ABC, D is a point on the side BC

In an equilateral triangle ABC, D is a point on the side BC such that BD = 1/3 BC. Prove that 9AD2 = 7AB2

### If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k - 1, 5k) are collinear

If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k - 1, 5k) are collinear, then find the value of k.

### Show that in a right triangle, the square of the hypotenuse is equal to

Show that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

### In given figure XY and X'Y' are two parallel tangents to a circle

In given figure XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. Prove that ∠ AOB = 90°. ### The points A(4, –2), B(7, 2), C(0, 9) and D(–3, 5) form a parallelogram

The points A(4, –2), B(7, 2), C(0, 9) and D(–3, 5) form a parallelogram. Find the length of the altitude of the parallelogram on the base AB.

### In what ratio does the x-axis divide the line segment joining the points (–4, –6) and (–1, 7)

In what ratio does the x-axis divide the line segment joining the points (–4, –6) and (–1, 7)? Find the co-ordinates of the point of division.

### If (1, p/3) is the mid-point of the line segment joining the points (2, 0) and (0, 2/9)

If (1, p/3) is the mid-point of the line segment joining the points (2, 0) and (0, 2/9), then show that the line 5x + 3y + 2 = 0 passes through the point (–1, 3p).