Question

In circle m triangle abc has been inscribed. It is known that ab=ac, DC is perpendicular to AB, and measurement of angle DCB is 23. Find: a) the measure of angle BDC b) the measure of arc DAC

Answer 1

a) ∠BDC = 46°

b) arc DAC = 222°

Explanation:

There may be shorter ways to work this, but here's one way.

We assume from the question regarding arc DAC that point D is on the circle (not on chord AB).

a) Call the point of intersection of CD and AB point X. Then triangle BXC is a right triangle, and angle CBX is the complement of angle DCB, hence 67°.

Triangle ABC is an isosceles triangle with angle ACB ≅ angle CBA, hence it, too, is 67°. Then angle ACD is the difference between 67° and the measure of angle DCB, so is 67° -23° = 44°.

The measure of arc AD will be double the measure of inscribed angle ACD, so is 2×44° = 88°. Because the angles at X are all 90°, arc BC is the supplement of arc AD, so is 180° -88° = 92°. This is double the measure of inscribed angle BDC, so ∠BDC = 92°/2 = 46°.

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b) Arc AC is double the measure of inscribed angle CBA, so is 2×67° = 134°. Arc DAC is the sum of the measures of arc AD (88°) and AC (134°), so is ...

arc DAC = 88° +134° = 222°