big ideas lines and segments that intersect circles

problem 1

<JKM = <JKL + <LKM

<JKM = 15° + 28°

<JKM = 43°

problem 2

AB = AC - BC

AB = 10 - 7

AB = 3

problem 3

chords intersect the circle at 2 points, but dont go through it

which means lines AD and BH are chords

problem 4

they will have 4 common tangents

see here

problem 5

the tangent is external

problem 6

tangents have a 90° angle to the radius, so we can use the pythagorean theorem to find if the triangle is a right triangle

15² + 9² ? 18²

225 + 81 ? 324

306 < 324

this means the triangle is not a right triangle

problem 7

we can use pythagorean theorem

r² + 24² = (r + 16)²

r² + 576 = r² + 32r + 256

576 = 32r + 256

32r = 320

r = 10

problem 8

r² + 30² = (r + 18)²

r² + 900 = 324 + 36r + r²

900 = 324 + 36r

36r = 576

r = 16

problem 9

3x² + 2x - 7 = 2x + 5

3x² - 7 = 5

3x² = 12

x² = 4

x = 2, x = -2

problem 10

the center of the circular pond is equidistant to the two bike paths because the points of tangency would be the same on both sides since the tangents started from the same point