Question

In the circle below, AD is a diameter and AB is tangent at A. suppose mADC=228. Find the measures of mCAB and mCAD. Type your numerical answers without units in each blank

Answer 1

m∠CAB = 66°

m∠CAD = 24°

Explanation:

m∠CAB

The given parameters are;

The measure of arc m
`\widehat{ADC}` = 228°

The diameter of the given circle =
`\overline{AD}`

The tangent to the circle =
`\underset{AB}{\leftrightarrow}`

The measure of m∠CAB and m∠CAD = Required

By the tangent and chord circle theorem, we have;

m∠CAB = (1/2) × m
`\widehat{AC}`

However, we have;

m
`\widehat{AC}` + m
`\widehat{ADC}` = 360° the sum of angles at the center of a circle is 360°

∴ m
`\widehat{AC}` = 360° - m
`\widehat{ADC}`

Which gives;

m
`\widehat{AC}` = 360° - 228° = 132°

m
`\widehat{AC}` = 132°

Therefore;

m∠CAB = (1/2) × 132° = 66°

m∠CAB = 66°

m∠CAD

Given that
`\overline{AD}` is the diameter of the given circle, we have

The tangent,
`\underset{AB}{\leftrightarrow}`, is perpendicular to the radius of the circle, and therefore
`\underset{AB}{\leftrightarrow}` is also perpendicular to the diameter of the circle

∴ m∠DAB = 90° which is the measure of the angle formed by two perpendicular lines

By angle addition property, we have;

m∠DAB = m∠CAB + m∠CAD

∴ m∠CAD = m∠DAB - m∠CAB

By substitution, we have;

m∠CAD = 90° - 66° = 24°

m∠CAD = 24°