Each solution below contains exactly one mistake. Circle the wrong line, explain what went wrong, then write the correct version.
①Solution 1
$A$, $B$ and $C$ are on a circle with centre $O$. The angle at the centre $AOC$ is $130$ degrees. Work out the angle $ABC$ at the circumference.
1
the angle at the centre is twice the angle at the circumference
2
so the angle at the circumference is $130 \times 2$
3
the angle $ABC$ is $260$ degrees
What went wrong?
Correct line:
②Solution 2
$PQRS$ is a cyclic quadrilateral. Angle $P$ is $85$ degrees. Work out angle $R$.
1
opposite angles in a cyclic quadrilateral are equal
2
$P$ and $R$ are opposite
3
so angle $R$ is $85$ degrees
What went wrong?
Correct line:
③Solution 3
$AB$ is a diameter of a circle and $C$ is on the circumference. Angle $CAB$ is $34$ degrees. Work out angle $ABC$.
1
the angle in a semicircle is $90$ degrees, so angle $ACB = 90$
2
the angles in a triangle add up to $180$
3
so angle $ABC = 180 - 90 - 34 = 66$ degrees
What went wrong?
Correct line:
④Solution 4
A tangent touches a circle at $T$, and $O$ is the centre. Work out the angle between the tangent and the radius $OT$.
1
a tangent just touches the circle at one point
2
it lies flat against the circle there, so it is parallel to the radius
3
the angle is $0$ degrees
What went wrong?
Correct line:
Answers · Circle Theorems
Spot the mistake
① Solution 1 line 2: the rule was quoted correctly and then applied backwards — the CENTRE angle is the doubled one, so the circumference angle is half of $130$. An angle of $260$ degrees inside a triangle is impossible anyway
Fix: the angle at the circumference is $130 \div 2$ $65\text{ degrees}$
② Solution 2 line 1: opposite angles in a cyclic quadrilateral ADD UP TO $180$ degrees — they are not equal unless both happen to be $90$
Fix: opposite angles in a cyclic quadrilateral add up to $180$ degrees $95\text{ degrees}$
③ Solution 3 line 3: the arithmetic is wrong: $180 - 90 - 34$ is $56$, not $66$
Fix: $180 - 90 - 34 = 56$ $56\text{ degrees}$
④ Solution 4 line 2: a tangent is PERPENDICULAR to the radius at the point of contact, not parallel to it
Fix: the tangent is perpendicular to the radius at the point of contact $90\text{ degrees}$
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