Find KN and LM. In triangle LKM, angle L is bisected. Line segment LK is 11, line segment KN is x-4, line segment NM is 5.4, and line segment LM is 2x + 1.3

(there is a drawing of this triangle)

Thanks.

I assume the bisector of angle L is LN where N is on KM

In the Post
http://www.jiskha.com/display.cgi?id=1359738761
I proved for you , and Steve called it, the angle bisector theorem
In this case:
KL/KN = LM/MN

11/(x-4) = (2x+1.3)/5.4
2x^2 - 6.7x - 5.2 = 59.4
2x^2 - 6.7x - 64.6 = 0
20x^2 - 67x - 646 = 0
x = (67 ± √56169)/40

= 7.6 or a negative which will not work in 2x+1.3

KN = x-4 = 7.6 - 4 = 3.6
LM = 2x+1.3 = 15.2 + 1.3 = 16.5

check:
LK/KN = 11/3.6 = 55/18
LM/MN = 16.5/5.4 = 55/18

Thanks.

To find the lengths KN and LM, we can use the fact that angle L is bisected. This means that the two angles formed at L are equal.

Let's start by finding the length of KN. Based on the given information, we know that LK = 11 and NM = 5.4. We need to find KN which is labeled as x-4.

Since angle L is bisected, we can set up the following equation:

LK/KN = LM/MN

Plugging in the given values:

11 / (x-4) = (2x + 1.3) / 5.4

Now we can cross multiply:

11 * 5.4 = (2x + 1.3) * (x - 4)

59.4 = 2x² - 8x + 1.3x - 5.2

Combining like terms:

2x² - 6.7x - 64.6 = 0

Now we have a quadratic equation in standard form. To solve it, we can use factoring, completing the square, or the quadratic formula.

In this case, the equation does not easily factor, so let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

For our quadratic equation, a = 2, b = -6.7, and c = -64.6:

x = (-(-6.7) ± √((-6.7)² - 4*2*(-64.6))) / (2*2)

Simplifying:

x = (6.7 ± √(44.89 + 515.2)) / 4

x = (6.7 ± √560.09) / 4

Now we calculate the two possible values of x:

x₁ = (6.7 + √560.09) / 4

x₁ = (6.7 + 23.66) / 4

x₁ = 30.36 / 4

x₁ ≈ 7.59

x₂ = (6.7 - √560.09) / 4

x₂ = (6.7 - 23.66) / 4

x₂ = -16.96 / 4

x₂ ≈ -4.24

Since the length of a segment cannot be negative, we discard x₂ ≈ -4.24 as an extraneous solution.

Therefore, KN ≈ 7.59 - 4 ≈ 3.59

To find the length of LM, we can plug the value of x into the equation: LM = 2x + 1.3

LM = 2 * 7.59 + 1.3

LM = 15.18 + 1.3

LM ≈ 16.48

So, KN ≈ 3.59 and LM ≈ 16.48.