Practice Questions

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

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 $\angle APB$.

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.

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

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

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

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.

Define a secant of a circle.

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.

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.

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.

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 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.

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

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

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.

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

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 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.

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.

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.

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.

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

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

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 $\triangle ABC$.

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.

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

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

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).

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.

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.

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.

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 $AB + CD + EF = BC + DE + FA$.