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Lesson 7 of 18

A corner standing on a diameter is a right angle

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8 min

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Nothing new is proved here. We put one number into the doubling rule: 180°.

A diameter AB, straight through the centre.
Step 1 of 4
  1. A diameter AB, straight through the centre.
  2. A point C, anywhere else on the circle.
  3. Triangle ABC, joined up.
  4. The corner at C, marked exactly 90°.

The rule

A diameter is a straight line through the centre. So it makes 180° at the centre. By the doubling rule, the edge angle is half of that. Half of 180° is 90°. Take a diameter AB and any other point C on the circle. Then angle ACB is exactly 90°, a right angle.

Exactly, not nearly

The 180° is not rounded. A diameter passes through the centre by definition. So the 90° is exact too.

Two near misses and the exact case, with the rim angle each gives
Angle at the centreAngle at the rimRight angle?
176°88°No
180° — a diameter90°Yes
184°92°No

A check with side lengths

In a triangle, the widest corner is opposite the longest side. Here the corner at C is a right angle. The other two corners total 90°, so each is smaller. Then the diameter opposite C is the longest side.

Move C round the circle. Then make AB shorter.

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