Question

Using the digits 0 to 9 at most one time

each, place a digit in each box so that the
central angle has the greatest possible
value.
central angle
measure =
inscribed angle
measure =
circumscribed
angle measure -

Answer 1

The required angles are;

Central angle measure is 176°

Inscribed angle measure is 88°

Circumscribed angle measure is 4°

Explanation:

Here we have;

Sum of angle at center + opposite circumscribed angle = 180

From circle theorem, angle at the center = 2 × Angle at circumference

Therefore, for maximum angle at the center, we have;

Angle at the center → 180°

Where angle at the center = 180°

Therefore, circumscribed angle = 180° - 180° = 0°

Again

2 × Angle at the circumference = angle at the center

∴ 2 × Angle at the circumference = 180°

∴ Angle at the circumference = 90°

Hence,

Central angle measure = 180°

Inscribed angle measure = 90°

Circumscribed angle measure = 0°

However, using the digit from 0 to 9 at most 1 time each, we have, counting down from 180° and comparing

Central angle measure = 176°

Inscribed angle measure = 88°

Circumscribed angle measure = 4°.