Question

Find the length of LM.

Write your answer as a whole number or a decimal

Answer 1

LM = 5

Explanation:

A secant is a straight line that intersects a circle at two points.

A segment is part of a line that connects two points.

Intersecting Secants Theorem

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.

The given diagram shows two secant segments LN and PN that intersect at exterior point N. Their external parts are MN and ON, respectively.

Therefore, according to the Intersecting Secants Theorem:

`\implies \sf LN \cdot MN =PN \cdot ON`

Given values:

• LN = 3 + x - 2 = x + 1
• MN = 3
• PN = 4 + x - 5 = x - 1
• ON = 4

Substitute the values into the equation and solve for x:

`\implies \sf LN \cdot MN =PN \cdot ON`

`\implies (x+1)\cdot 3=(x-1)\cdot 4`

`\implies 3x+3=4x-4`

`\implies 4x-4=3x+3`

`\implies 4x-4-3x=3x+3-3x`

`\implies x-4=3`

`\implies x-4+4=3+4`

`\implies x=7`

To determine the length of LM, substitute the found value of x into the expression for the line segment:

`\implies \overline{LM}=7-2=5`

Therefore, the length of LM is 5.