In the following figure, in ΔPQR, seg RS is the bisector of ∠PRQ. If PS = 6, SQ = 8, PR = 15, find QR.

Concept: Similarity of Triangles

If two circles with radii 8 cm and 3 cm, respectively, touch internally, then find the distance between their centers.

Concept: Touching Circles

In Fig. 3, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and ∠APB = 60°. Find the length of chord AB.

Concept: Areas of Sector and Segment of a Circle

A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely?

Concept: Surface Area of a Combination of Solids

In the following figure seg AB ⊥ seg BC, seg DC ⊥ seg BC. If AB = 2 and DC = 3, find `(A(triangleABC))/(A(triangleDCB))`

Concept: Properties of Ratios of Areas of Two Triangles

Prove that the angle bisector of a triangle divides the side opposite to the angle in the ratio of the remaining sides.

Concept: Similarity of Triangles

In the following figure, seg BE ⊥ seg AB and seg BA ⊥ seg AD. If BE = 6 and \[\text{AD} = 9 \text{ find} \frac{A\left( \Delta ABE \right)}{A\left( \Delta BAD \right)} \cdot\]

Concept: Similarity of Triangles

In the given figure, AD is the bisector of the exterior ∠A of ∆ABC. Seg AD intersects the side BC produced in D. Prove that :

Concept: Properties of Ratios of Areas of Two Triangles

Find the height of an equilateral triangle whose side is 6 units.

Concept: 30 - 60 - 90 and 45 - 45 - 90 Theorem

In a right angled triangle, if sum of the squares of the sides making right angle is 169 then what is the length of the hypotenuse?

Concept: Apollonius Theorem

Prove that “The lengths of the two tangent segments to a circle drawn from an external point are equal.”

Concept: Number of Tangents from a Point on a Circle

If two circles with radii 5 cm and 3 cm respectively touch internally, find the distance between their centres.

Concept: Touching Circles

Prove that ‘the opposite angles of a cyclic quadrilateral are supplementary’.

Concept: Cyclic Quadrilateral

Find the circumferences of a circle whose radius is 7 cm.

Concept: Tangents and Its Properties

Prove that “The opposite angles of a cyclic quadrilateral are supplementary”.

Concept: Cyclic Properties

**Choose the correct alternative.**

∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB).

Concept: Inscribed Angle

A (5, 4), B (-3, -2) and C (1, -8) are the vertices of a triangle ABC. Find the equations of the median AD and line parallel to AC passing through the point B.

Concept: General Equation of a Line

Find the slope and y-intercept of the line y = -2x + 3.

Concept: Intercepts Made by a Line

Find the slope and *y* – intercept of the line *y* = 3*x* – 5.

Concept: Intercepts Made by a Line

A (5, 4), B (–3,–2) and C (1,–8) are the vertices of a triangle ABC. Find the equation of median AD and line parallel to AB passing through point C.

Concept: General Equation of a Line