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

In the following figure RP: PK= 3:2, then find the value of A(ΔTRP):A(ΔTPK).

Concept: Properties of Ratios of Areas of Two Triangles

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

Concept: Cyclic Quadrilateral

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

Concept: Touching Circles

A person standing on the bank of river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Find the height of the tree and width of the river. `(sqrt 3=1.73)`

Concept: Heights and Distances

Using Euler’s formula, find V if E = 30, F = 12.

Concept: Euler's Formula

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

**Some question and their alternative answer are given.**

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

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

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

Concept: Touching Circles

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

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