Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) What is the locus of all points from which two tangents drawn to a circle are equal in length? All points the circle (or any point the circle)

Perfect! Every external point has equal tangent lengths to a circle.

(2) State the converse of the theorem: "The tangent to a circle is perpendicular to the radius at the point of contact." If a is to a radius at a point the circle, then it is a to the circle

Excellent! This is how we can construct tangents using perpendiculars.

(3) If two tangents are drawn from an external point to a circle and one of them makes an angle of 35° with the line joining the external point to the center, find the angle between the tangents. Angle between tangents = °

Correct! The external point forms an isosceles triangle with equal tangent lengths.

(4) State True or False: "A secant to a circle can be perpendicular to the radius."

Correct! A secant can intersect the circle at any angle, including perpendicular to a radius.

(5) Name the quadrilateral formed by joining the center of the circle and the two points of contact of tangents from an external point.

Perfect! It has two pairs of adjacent equal sides.

(6) Is the triangle formed by two radii and a chord always isosceles? Justify. ,because both have lengths.

Excellent! All radii of a circle are equal by definition.

(7) Can a tangent intersect the center of the circle? Explain. ,because a tangent touches the circle at exactly point and the center is the circle.

Perfect understanding! Tangents only touch the circumference.

(8) What will be the angle between the tangents if the radius is 6 cm and the distance of the external point from the center is 6 cm? °

Since distance = radius, the point is the circle, so only tangent exists. Angle is undefined (or 0°)

Excellent analysis! When distance equals radius, the point is on the circle.

Short Answer Questions (2 Marks Each)

Note: Answer each question with steps and explanation, in 2-3 sentences. Write down the answers on sheet and submit to the school subject teacher.

(1) Two tangents TP and TQ are drawn to a circle from an external point T. Prove that triangle △TPQ is isosceles and also find ∠PTQ if radius is 5 cm and OT = 13 cm.

(2) A point is located outside a circle. From this point, a tangent of length 15 cm is drawn to the circle. If the radius is 9 cm, find the distance of the point from the center of the circle. Distance = cm

Perfect application of the tangent-radius relationship!

(3) Construct a circle and draw a chord AB. Construct a tangent to the circle at point A using ruler and compass.

(4) A circle of radius 10 cm has a chord of length 12 cm. Find the length of the tangent drawn from the midpoint of the chord. cm.

Mid-point is inside, so no external tangent exists! Internal points cannot have external tangents.

(5) Prove that the quadrilateral formed by joining two radii and two tangents from an external point is cyclic.

(6) Find the area of the triangle formed by two tangents drawn from an external point to a circle and the line joining the point to the center, if radius = 6 cm and length of tangent = 8 cm. Area = cm2

Excellent trigonometric calculation!

(7) Prove that the angle subtended by the chord at the center is twice the angle subtended on the circumference using the properties of tangents.

Long Answer Questions (4 Marks Each)

Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.

(1) Prove that the tangents drawn from an external point to a circle are equal and the angle between the tangents is supplementary to the angle subtended by the chord joining the points of contact at the center.

(2) Two circles of radii 4 cm and 1 cm touch each other externally. Construct a common external tangent to both circles and measure its length. Prove your construction.

(3) A triangle is inscribed in a circle such that one side is a diameter. Prove that the tangent drawn at one endpoint of the diameter is perpendicular to the opposite vertex of the triangle.

(4) Two tangents TP and TQ are drawn to a circle with center O such that ∠POQ = 120°. Find ∠PTQ and the area of triangle △PTQ if OP = OQ = 6 cm.

∠PTQ = ° and Area of △PTQ = cm2

Excellent trigonometric calculation!

(5) Let the length of the tangent from an external point P to a circle with center O and radius r be l. Prove that triangle △POT is a right-angled triangle and express r, l, and OP in terms of Pythagorean relation. Then solve for r = 5 cm, l = 12 cm.

△POT is right-angled at with OP = cm

Perfect application of Pythagorean theorem to tangent problems!

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) The locus of all points from which equal tangents can be drawn to a circle is:

(a) Diameter (b) Radius (c) Circle (d) Straight Line

Diameter

Radius

Circle

Straight Line

Correct!

(2) If ∠POQ = 60°, where P and Q are points of contact of tangents from external point T, then ∠PTQ is:

(a) 120° (b) 60° (c) 90° (d) 45°

120°

60°

90°

45°

Correct! ∠POQ + ∠PTQ = 180° (supplementary), so ∠PTQ = 180° - 60° = 120°.

(3) The triangle formed by two tangents and the line joining the center to the external point is:

(a) Isosceles and Right (b) Equilateral (c) Scalene (d) None

Isosceles and Right

Equilateral

Scalene

None

Correct! It's isosceles (equal tangents) and has right angles where tangents meet radii.

(4) If the distance from the center to the external point is d and radius is r, then the length of the tangent is:

(a) r2−d2 (b) d2−r2 (c) r + d (d) r – d

√(r² – d²)

√(d² – r²)

r + d

r – d

Correct! Using Pythagorean theorem: d2 = r2 + l2, so l = d2−r2.

(5) If a tangent at point A and a chord AB are drawn in a circle such that ∠CAB = 35° , where C is a point on the opposite arc, then ∠ACB is:

(a) 55° (b) 35° (c) 145° (d) 90°

55°

35°

145°

90°

Correct!

(6) The angle between two tangents drawn from an external point increases when the external point:

(a) Moves closer to the circle (b) Moves away from the circle (c) Moves around the circle (d) Is at the center

Moves closer to the circle

Moves away from the circle

Moves around the circle

Is at the center

Correct! As the external point approaches the circle, the angle between tangents increases.

(7) If ∠PTQ = 70°, then ∠POQ = ?

(a) 110° (b) 70° (c) 90° (d) 140°

110°

70°

90°

140°

Correct! ∠PTQ + ∠POQ = 180°, so ∠POQ = 180° - 70° = 110°.

(8) A tangent from a point A to a circle of radius r and center O has length l. Then in △OAP:

(a) r2= OA2 - OP2 (b) OP2 = OA2 + r2 (c) OA2 = OP2 – r2 (d) OP2 = OA2 – r2

r² = OA² - OP²

OP² = OA² + r²

OA² = OP² – r²

OP² = OA² – r²

That's correct!

(9) Two tangents are drawn from an external point to a circle. The angle between the tangents is 90°. What is the ratio of the radius to the distance from the point to the center?

(a) 1 : 1 (b) 1 : 2 (c) 1 : 2 (d) 2 : 1

1 : 1

1 : √2

1 : 2

2 : 1

Correct! When angle between tangents is 90°, the triangle formed has special properties giving this ratio.

(10) The length of a tangent from a point P to a circle is maximum when the radius is:

(a) Zero (b) Equal to the distance from P to the center (c) Minimum (d) Perpendicular to the line joining P and center

Zero

Equal to the distance from P to the center

Minimum

Perpendicular to the line joining P and center

Correct! Since tangent length = d2−r2, it's maximum when r is minimum (for fixed d).

One tangent only

No external tangents

√(d² - r²)

Two equal tangents

Perpendicular to radius

Not applicable

External Point

Point on Circle

Internal Point

Tangents and Secants Challenge

Determine whether these statements about tangents and secants are True or False:

If a chord and a tangent intersect outside the circle, the tangent length is always greater than either segment of the secant.

If two circles intersect each other, their common tangents can be two, three, or four in number depending on their positions.

For any external point, the distance from the point to the center of the circle is less than the length of the tangent from that point.

If two tangents are drawn from an external point to a circle, then the line joining the external point to the center of the circle bisects the angle between the tangents.

The product of the lengths of two segments of a secant from an external point is equal to the square of the length of the tangent drawn from the same point.

The locus of points from which tangents of equal length can be drawn to a circle is the diameter of the circle.

Tangents and Secants Quiz

🎉 You Did It! What You've Learned:

By completing this worksheet, you now have a solid understanding of:

(1) Tangent Properties: Perpendicularity to radius and equal lengths from external points

(2) Angle Relationships: Supplementary angles between tangent pairs and central angles

(3) Pythagorean Applications: Finding distances and lengths in tangent problems

(4) Construction Methods: Drawing tangents using compass and ruler techniques

(5) Quadrilateral Properties: Understanding kites and cyclic quadrilaterals formed by tangents

(6) Geometric Proofs: Proving equality and angle relationships using congruence

(7) Real-world Applications: Calculating tangent lengths and solving practical problems

(8) Advanced Concepts: Common tangents to two circles and complex geometric relationships

Excellent work mastering the advanced concepts of tangents and secants to circles!

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