Tangents to a Circle Practice

Points A and B are contact points of tangents drawn from external point P to a circle. If PA = 14 cm, what is the length of PB?

14 cm, because tangents drawn from a single external point to a circle are equal in length.

A circle has a radius of 5 cm. A point P lies 13 cm away from the center O. What is the length of the tangent segment from P to contact point A?

12 cm. Triangle OAP is a right triangle with right angle at A. By the Pythagorean theorem: PA = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12 cm.

Two tangents PA and PB meet a circle at A and B. The central angle AOB measures 110 degrees. Find the measure of angle APB.

70 degrees. Quadrilateral OAPB has interior angles summing to 360 degrees. Angles OAP and OBP are 90 degrees each (totaling 180 degrees), so angle APB = 180 - 110 = 70 degrees.

Tangents PA and PB form an angle APB of 40 degrees. What is the measure of angle PAB in triangle PAB?

70 degrees. Since PA = PB, triangle PAB is isosceles with angle PAB equal to angle PBA. The remaining angle sum is 180 - 40 = 140 degrees, so angle PAB = 140 / 2 = 70 degrees.

A tangent meets chord AB at point A on a circle. The acute angle between the tangent and chord AB is 55 degrees. Point C is on the major arc of the circle. What is the measure of the inscribed angle ACB?

55 degrees, by the alternate segment theorem (the angle between a tangent and a chord equals the angle subtended by the chord in the alternate segment).

A quadrilateral ABCD is circumscribed around a circle so that all four sides are tangent to the circle. If AB = 8 cm, BC = 11 cm, and CD = 9 cm, find the length of side DA.

6 cm. In any tangential quadrilateral, the sums of opposite sides are equal: AB + CD = BC + DA. Substituting gives 8 + 9 = 11 + DA, so 17 = 11 + DA, meaning DA = 6 cm.