Geometry (w/ factoring)

Q:  Geometry and Angles

Ray RY bisects angle NRT
Measure of angle NRY = 2x² + 6x
Measure of angle YRT = 10x + 6
Find x and the measure of angle YRT

A:  Ahhh, fun to answer without a picture.  But, we can do it.  We have an angle that is bisecting.  Bisected means cut in half.  Therefore, angle NRY = angle YRT:

Mathematically:  2x² + 6x = 10x + 6

Now, time to solve!  Since I see an “x²”, I think to myself that we must factor this.  First step to factoring?  Set the equation equal to zero:

2x² + 6x = 10x + 6

2x² + 6x – 10x = 6

2x² + 6x – 10x – 6 = 0

2x² – 4x – 6 = 0

Now, this is a quadratic that needs solving.  We can either solve using the quadratic equation or by factoring.  Let’s factor (View Factoring 101 for the basics of factoring):

2x² – 4x – 6 = 0

(2x          )(x          ) = 0

(2x  +      )(x  –      ) = 0  [I know there is a – and a + because the two numbers must multiple to give -6… And, a – times a + equals a -, right?  However, I do not know that the + goes with the 2x and the – goes with the x… This is just my best guess to start.  Factoring can be treated as a guessing game – if you use guess and check, you get better at it with time.]

(2x + 2)(x – 3) = 0

So,

(2x + 2) = 0 or (x – 3) = 0

x = -1 or 3

We have two potential answers.  Take note, if x=-1, then we have:

Angle NRY = 2x² + 6x = 2(-1)² + 6(-1) = 2 – 6 = -4

An angle cannot be negative, therefore x cannot be -1.

Therefore, x must equal 3:

Angle YRT = 10x + 6

So, if x = 3:

Angle YRT = 10(3) + 6 = 30 + 6 = 36.