Practice Questions

How many tangents can be drawn to a circle from a point lying inside it?

What is the name of the common point between a tangent and a circle?

List the number of tangents that can be drawn to a circle from a point based on its position. Identify the three possible positions for the point.

Two concentric circles have radii of 13 cm and 5 cm. Formulate a method to find the length of a chord of the larger circle that is tangent to the smaller circle and justify your steps.

A quadrilateral PQRS is drawn to circumscribe a circle. If $PQ = 12$ cm, $QR = 15$ cm, and $RS = 14$ cm, calculate the length of SP.

A point Q is 26 cm away from the centre of a circle and the length of the tangent drawn from Q to the circle is 24 cm. Calculate the radius of the circle.

Two tangents PA and PB are drawn from an external point P to a circle with centre O. If $\angle AOB = 130^\circ$, calculate the measure of $\angle APB$.

Justify the statement: 'A circle can have an infinite number of tangents.'

Define a secant of a circle.

Fill in the blank: The tangent at any point of a circle is ________ to the radius through the point of contact.

State Theorem 10.1 regarding the relationship between a tangent and the radius of a circle. Describe a diagram that illustrates the theorem.

Define the following terms related to a circle: (i) Tangent, (ii) Chord, (iii) Point of Contact.

How many tangents, parallel to a given secant, can a circle have at the most?

Describe the relationship between a secant and a tangent. Explain how a tangent can be considered a special case of a secant.

State Theorem 10.2 about the lengths of tangents from an external point. Describe a diagram that illustrates this property.

Justify why the center of a circle must lie on the angle bisector of the angle formed by two tangents drawn from an external point.

Explain the three possible relationships between a line and a circle in a plane. Use a simple description for each case.

Summarize the key properties of tangents to a circle. List at least five distinct properties or theorems.

Explain the concept of 'length of the tangent' from an external point to a circle. Use a diagram description to show two tangents from an external point P to a circle with centre O and points of contact T₁ and T₂. Identify the lengths of the tangents and state the relationship between them.

A circle touches the sides of $\triangle PQR$ at points A, B, and C on sides PQ, QR, and RP respectively. If $PQ = 12$ cm, $QR = 8$ cm, and $RP = 10$ cm, calculate the length of PA.

Two concentric circles have radii of 13 cm and 5 cm. Calculate the length of the chord of the larger circle that is tangent to the smaller circle.

A circle with center O has a radius of 5 cm. A point P is located 13 cm from O. Two tangents, PQ and PR, are drawn from P to the circle. Calculate the area of the quadrilateral PQOR.

A circle is inscribed in a $\triangle ABC$ having sides $AB = 8$ cm, $BC = 10$ cm, and $AC = 12$ cm. The circle touches the sides at points D, E, and F on AB, BC, and AC respectively. Calculate the lengths of AD, BE, and CF.

From an external point P, two tangents PA and PB are drawn to a circle with centre O and radius $r$. If $\angle APB = 60^\circ$, analyze the geometry to express the length of the tangent PA in terms of the radius $r$.

Two tangents are drawn to a circle from an external point P, touching the circle at A and B. The circle has a radius of 9 cm and the centre O is 15 cm away from P. Calculate the length of the chord AB.

A circle is inscribed in a right-angled triangle whose legs (sides containing the right angle) are 8 cm and 15 cm. Calculate the radius of the inscribed circle.

A circle touches the side BC of a $\triangle ABC$ at point P, and touches the extended sides AB and AC at points Q and R respectively. Analyze the lengths of the tangents from vertices A, B, and C to demonstrate that the length of the tangent AQ is half the perimeter of $\triangle ABC$.

Evaluate why it is impossible to construct a tangent to a circle from a point located inside it.

Justify that the tangents drawn at the endpoints of a diameter of a circle are parallel.

A circle is inscribed in a $\triangle ABC$ with sides $AB = 12$ cm, $BC = 8$ cm, and $AC = 10$ cm. The circle touches the sides at D, E, and F respectively. Evaluate the lengths of AD, BE, and CF.

Formulate a formal proof for the theorem stating that the angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the center.

Explain with the help of a diagram description why it is not possible to draw a tangent to a circle from a point inside it, but it is possible to draw exactly two tangents from a point outside it.

In a right-angled triangle, create a proof to show that the radius 'r' of its incircle is given by the formula $r = \frac{a + b - c}{2}$, where a and b are the lengths of the legs and c is the length of the hypotenuse.

Describe the logical steps used to explain why the tangent at any point of a circle is perpendicular to the radius through the point of contact (Theorem 10.1).

Critique the following statement: 'A secant is a generalized form of a tangent.'

A circle with center O is inscribed in a quadrilateral ABCD. Formulate a proof that the opposite sides of the quadrilateral subtend supplementary angles at the center of the circle. That is, prove $\angle AOB + \angle COD = 180^\circ$.

Formulate a proof for this generalization of Pitot's theorem: 'If a hexagon ABCDEF circumscribes a circle, the sum of the lengths of its alternate sides are equal.' That is, prove $AB + CD + EF = BC + DE + FA$.

Critique the following flawed proof of Theorem 10.1 (The tangent at any point of a circle is perpendicular to the radius through the point of contact). Identify the logical fallacy.

Flawed Proof: 'Let XY be the tangent at point P to a circle with center O. Join OP. In $\triangle OPX$, $\angle OPX$ must be $90^\circ$ because the tangent and radius meet. Therefore, $OP \perp XY$.'

Design a proof to show that if two tangents are drawn to a circle from an external point, then they are equally inclined to the line segment joining the center to that point.

From a point P, two tangents PA and PB are drawn to a circle with center O. If the length of OP is equal to the diameter of the circle, justify that $\triangle APB$ is an equilateral triangle.

PQ is a chord of length 24 cm in a circle with a radius of 13 cm. Tangents drawn at points P and Q intersect at an external point T. Calculate the length of the tangent TP.

Formulate a concise argument to prove that a parallelogram circumscribing a circle must be a rhombus.