Question

In triangle `ABC`, `AB = x`, `BC = 5" cm"`, `AC = (10-x)" cm"`, and `cos (angle ABC) = -1/7`. What is the value of `x`?

Answer 1

`x = 3.5" cm"`

Explanation:

Given: `AB = x" cm"`, `BC = 5" cm"`, `AC = (10-x)" cm"`, and `cos(angle ABC) = -1/7`

I will switch to the notation where the angle is represented by an uppercase letter and the opposite side is represented by the corresponding lowercase letter:

`c = x" cm"`, `a = 5" cm"`, `b = (10-x)" cm"`, and `cos(B) = -1/7`

We may use the Law of Cosines to derive an equation that has x as the only variable:

`b^2 = a^2+c^2-2(a)(c)cos(B)`

Substitute in the known values:

`((10-x)" cm")^2 = (5" cm")^2+x^2-2(5" cm")(x)(-1/7)`

Expand the square and write both sides in standard form:

`x^2 - 20 x + 100 = x^2 + (10 x)/7 + 25`

Solve for x:

`-20 x + 100 = (10 x)/7 + 25`

`-150/7x=-75`

`x = 3.5" cm"`