grade 9 Circle and Line

Circle and Line

A line meets a circle in one of three ways. If the center is closer than the radius, it cuts through at two points; at the radius it just touches as a tangent; farther out, they never meet. A tangent meets the radius at a right angle, and two tangents from an outside point are equal. Change the distance and a secant becomes a tangent, then misses the circle.

Circle vs Line — Position

Distance d80

📏 Three Cases

①d < r → meets at 2 points (secant)

②d = r → meets at 1 point (tangent)

③d > r → no intersection

④Chord length = 2√(r² − d²). Here r is 80 in the same units as the distance slider. Default distance 80 is the tangent. Power of a point and tangential quadrilaterals are outside this chapter

Properties of a Tangent

Position of P (mark on the circle)20

✨ Key Tangent Facts

①Tangent ⊥ radius at the contact point (OT ⊥ PT)

②Two tangents from one external point are equal (PA = PB)

③Tangent length = √(OP² − r²) (Pythagoras)

Perpendicular Bisector of a Chord

Chord Property

Perpendicular from center to chord → bisects it

Center → perpendicular = perpendicular bisector of the chord

Chord Length

Chord = 2√(r² − d²)

d = distance from center to chord

💡 Chord Theorems

①Equal-distance chords have equal lengths

②The longer chord is closer to the center

③Longest chord = diameter (d = 0)

Tangent-Chord Angle

Tangent-chord angle = inscribed angle on the chord

Tangent-chord angle theorem

📐 Using the Tangent-Chord Theorem

①Tangent-chord angle equals the inscribed angle on the other side

②Common test setup

③The angle equals the inscribed angle on the opposite arc

Work It Out

Example 1

In a circle of radius 5, find the length of a chord whose distance from the center is 3.

Chord length = 2√(r²−d²), with r=5, d=3.

chord = 2√(5²−3²)

= 2√16 = 2·4 = 8.

= 2√16 = 8

▸ 8

The perpendicular from the center bisects the chord → right triangle + Pythagoras.

Example 2

From an external point P, find the length of the tangent to the circle. (OP=13, radius r=5)

Tangent length = √(OP²−r²), with OP=13, r=5.

tangent = √(13²−5²)

= √(169−25) = √144 = 12.

= √144 = 12

▸ 12

Tangent ⊥ radius gives a right triangle → Pythagoras.

Exam Wrap-up

Core

Tangent ⊥ radius, PA = PB, chord = 2√(r² − d²)

Memorize these three

G9 school-exam type

From an external point P, two tangents touch the circle at A and B. If ∠APB=50°, find ∠AOB (O is the center).

①110°

②120°

③130°

④140°

⑤150°

▸ ③ 130°

Since tangent ⊥ radius, ∠OAP=∠OBP=90°.

∠OAP = ∠OBP = 90°

In quadrilateral PAOB (angle sum 360°), ∠AOB = 360−90−90−50 = 130°.

∠AOB = 360−90−90−50 = 130°

🎯 Exam Points

①Tangent ⊥ radius → right triangle → Pythagoras

②Two tangents from one external point are equal: PA=PB

③Perpendicular bisector of a chord passes through the center

④Secant-tangent: use the power of a point

⑤Tangential quadrilateral: opposite sides sum equal

← Previous

Trigonometric Ratios

Inscribed Angle

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