Flashcards · 13 cards

Circle geometry

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Rule
An angle at the centre and an angle at the circumference stand on the same arc. How are they related?
Answer
The angle at the centre is twice the angle at the circumference: $\angle AOB = 2 \times \angle APB$ .
Angle at the centre∠AOB = 2 × ∠APB, on the same arc
From the lesson: The angle at the centre
Rule
Two angles stand on the same chord, in the same segment. What do you know about them?
Answer
They are equal: $\angle APB = \angle AQB$ .
Same segment∠APB = ∠AQB
From the lesson: Same segment and semicircle
Rule
What is the angle in a semicircle?
Answer
$90^\circ$ . If $AB$ is a diameter, $\angle APB = 90^\circ$ wherever $P$ is on the circle.
Angle in a semicircleAB a diameter, so ∠APB = 90°
From the lesson: Same segment and semicircle
Rule
The opposite angles of a cyclic quadrilateral add up to what?
Answer
$180^\circ$ : $\angle A + \angle C = 180^\circ$ , and $\angle B + \angle D = 180^\circ$ .
Opposite angles∠A + ∠C = 180°
From the lesson: Cyclic quadrilaterals
Rule
The exterior angle of a cyclic quadrilateral is equal to which angle?
Answer
The interior opposite angle: $\angle DCE = \angle A$ .
Exterior angle∠DCE = ∠A
From the lesson: Cyclic quadrilaterals
Rule
A tangent touches the circle at $T$ . What angle does it make with the radius $OT$ ?
Answer
$90^\circ$ : $\angle OTP = 90^\circ$ .
Tangent and radius∠OTP = 90°
From the lesson: Tangents
Rule
Two tangents are drawn to a circle from a point $P$ outside it. What is true of their lengths?
Answer
They are equal: $PT_1 = PT_2$ .
Two tangentsPT₁ = PT₂
From the lesson: Tangents
Rule
What does the perpendicular from the centre to a chord do to the chord?
Answer
It bisects it: $AM = MB$ . With the radius $OB$ that makes a right-angled triangle, so $OB^2 = OM^2 + MB^2$ .
Perpendicular from the centreOM ⊥ AB, so AM = MB and OB² = OM² + MB²
From the lesson: Tangents
Rule
The angle between a tangent and a chord is equal to which angle?
Answer
The angle in the alternate segment: $\angle ATS = \angle TXA$ .
Alternate segment∠ATS = ∠TXA
From the lesson: Tangents
Rule
Two chords $AB$ and $CD$ cross at $X$ inside a circle. What is equal?
Answer
$AX \times XB = CX \times XD$ : the two parts of one chord multiply to the same as the other's.
Crossing chordsAX × XB = CX × XD
From the lesson: Tangents
Rule
From a point $P$ outside a circle, a tangent touches it at $T$ and a line cuts it at $A$ and $B$ . What is the rule?
Answer
$PT^2 = PA \times PB$ , with both lengths measured from $P$ .
Tangent and secantPT² = PA × PB, both measured from P
From the lesson: Tangents
Which method?
WAEC 2020 · Paper 1 · Q33
In the diagram, $O$ is the centre of the circle and $RP$ is a diameter. If $\angle OPQ = 48^\circ$ , find the value of $m$ .
What do you notice first?
Answer
$OP$ and $OQ$ are both radii, so triangle $OPQ$ is isosceles and its base angles are equal. Then use the straight line $RP$ .
From the lesson: The angle at the centre Try the question
Which method?
JAMB 1994 · UME · Q27
In the diagram, $O$ is the centre of the circle and $SOQ$ is a diameter. If $\angle PRS = 38^\circ$ , what is the value of $\angle PSQ$ ?
Which two facts unlock this?
Answer
$SOQ$ is a diameter, so $\angle SPQ = 90^\circ$ (the angle in a semicircle). $\angle PQS$ and $\angle PRS$ stand on $PS$ in the same segment, so they are equal. Then the angles of triangle $PSQ$ add to $180^\circ$ .
From the lesson: Same segment and semicircle Try the question