triangle ABC is inscribed in the circle and AC = AB. The measure of angle BAC is 42 degrees and segment ED is tangent to the circle at point C. What is the measure of angle ACD?
If you draw a circle, locate the centre, then the inscribed triangle is isosceles.
Join the centre to all 3 triangle vertices.
The angle at A is now split into 2 equal 21 degree angles.
If we label the centre F, then angle FAB = angle FAC = 21 degrees.
The 3 triangles within the inscribed triangle ABC are all isosceles also.
Hence, angle FCA = angle FBA = 21 degrees.
The tangent is perpendicular to the line CF.
Therefore the sum of angles FCA and ACD is 90 degrees.
Hence angle ACD is 90-21 = 69 degrees.
Hello, sri340!
Great solution, Archie!
Here's another approach . . .
Triangle is inscribed in the circle and
The measure of is 42°
. . and segment is tangent to the circle at point
What is the measure of ?Code:A * o * * / \ * 138° * /42°\ * * / \ * / \ o D * / \ * / * / \ * / * / \ */ / 69° \ / B o- - - - - - - - -o C * * * */ * * * / / o E
Since is isosceles,
Then
. . An inscribed angle is measured by one-half its intercepted arc.
Therefore: .
. . The angle between a common tangent and secant
. . . . is measured by one-half its intercepted arc.