4 mistakes learners make with circle theorems, each one named and explained.
Every circle theorem has a condition attached, and the condition is the part that decides whether it applies.
You mark the two halves of a chord equal because a radius crosses it, at any angle.
The radius comes from the centre, and anything from the centre feels symmetrical.
Question. A radius crosses a chord. Does it cut the chord in half?
A common answer. yes, because it comes from the centre
The answer. only if it meets the chord at a right angle
Why. The perpendicular from the centre bisects the chord; a radius at any other angle does not.
Look for the right angle mark. Without it there is no bisection.
You mark two angles as equal because they stand on the same chord, with one above the chord and one below it.
The chord is the thing you can see, and the segment it divides the circle into is not drawn in.
Question. Two angles stand on the same chord, one in each segment. Are they equal?
A common answer. yes, same chord means same angle
The answer. no. They are equal only in the same segment, and here they add to 180 degrees
Why. Across the chord the two angles are the opposite angles of a cyclic quadrilateral.
Shade the segment. Equal angles have to be standing in the same shaded piece.
You mark a right angle in a triangle whose three vertices are on the circle, with no diameter anywhere.
The theorem is memorable and the picture looks the same, so the condition about the diameter drops away.
Question. A triangle has all three vertices on a circle. Is it right-angled?
A common answer. yes, triangles in a circle are right-angled
The answer. only if one of its sides is a diameter
Why. The right angle comes from the semicircle, so without a diameter there is no semicircle.
Find the diameter first. No diameter, no right angle.
A shape has three vertices on the circle and one at the centre, and you treat the angle at the centre as the one that adds to 180.
Both rules are about four points and a circle, and the picture is nearly the same.
Question. A quadrilateral has three vertices on the circle and the fourth at the centre. Which rule applies?
A common answer. opposite angles add to 180 degrees
The answer. the angle at the centre is twice the angle at the circumference
Why. Opposite angles adding to 180 needs all four vertices on the circle.
Count how many vertices are actually on the circle before picking the rule.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.