Question

Find KL and show work

Answer 1

Let's see

• LM(KL+LM)=NM(NM+NS)
• 6(6+x+15)=7(7+11)
• 6(x+21)=7(18)
• 6(x+21)=126
• x+21=21
• x=0

So

• KL=0+15=15

Answer 2

Secant: a straight line that intersects a circle at two points.

Intersecting Secants Theorem

If two secant segments are drawn to the circle from one exterior point, the product of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and its external part.

From inspection of the given diagram:

• M = Exterior point
• MK = secant segment and ML is its external part
• MS = secant segment and MN is its external part

Therefore:

⇒ ML · MK = MN · MS

Given:

• MK = (x + 15) + 6 = x + 21
• ML = 6
• MS = 7 + 11 = 18
• MN = 7

Substituting the given values into the formula and solving for x:

⇒ ML · MK = MN · MS

⇒ 6(x + 21) = 7 · 18

⇒ 6x + 126 = 126

⇒ 6x = 0

⇒ x = 0

Substituting the found value of x into the expression for KL:

⇒ KL = x + 15

⇒ KL = 0 + 15

⇒ KL = 15