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

Answer:

m∠CAB = 66°

m∠CAD = 24°

Step-by-step 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°