Tangents to Circles

More Pythagorean Theorem type problems! Yeah!!  Point of Tangency Theorem Point of Tangency More Pythagorean Theorem type problems! Yeah!!  If a line (segment or ray) is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

1. Find x A 12 B 9 a2 + b2 = c2 x 92 + 122 = x2 x = 15

RQ = 16 2. Find RQ a2 + b2 = c2 P 12 8 R Q 122 + (QR)2 = (8+12)2

r = 10 r2 + 242 = (r + 16)2 3. Find the radius. 16 A C 24 B

Point of Tangency Theorem (Converse) If a line is perpendicular to the radius at its endpoint, then the line is tangent to the circle

Determine if LM is tangent to the circle. To show that LMN is a right triangle we use the Pythagorean Theorem: N M L 6 cm 8 cm 10 cm The lengths of the sides of the triangle satisfy the Pythagorean Theorem, so LM is perpendicular to MN and is therefore a tangent to the circle.

Theorem about the Intersection of two tangent line segment If two tangent lines intersect at one point, the segments from the point to the point of tangency are congruent.

S If two segments from the same exterior point are tangent to a circle, then they are congruent. R T Party hat problems!

4. Find x R S T

5. Find x C A B

6. Find x. B A C P D E

7. Find NP N T S P R Q