Question

measure of the angle for the 5 ball
measure of the angle for the 9 ball
measure of the angle for 4 ball
between which two balls is the distance the greatest?

Explanation:

Step1: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is 180 degrees. So, \((4x - 1)^{\circ}+(3x - 5)^{\circ}+(8x + 2)^{\circ}=180^{\circ}\).

Step2: Combine like - terms

\((4x+3x + 8x)+(-1-5 + 2)=180\), which simplifies to \(15x-4 = 180\).

Step3: Solve for x

Add 4 to both sides: \(15x=180 + 4=184\), then \(x=\frac{184}{15}\approx12.27\).

Step4: Find the measure of each angle

For the angle corresponding to the 5 - ball (\(4x - 1\)):
Substitute \(x=\frac{184}{15}\) into \(4x - 1\), \(4\times\frac{184}{15}-1=\frac{736}{15}-1=\frac{736 - 15}{15}=\frac{721}{15}\approx48.07\) degrees.
For the angle corresponding to the 9 - ball (\(3x - 5\)):
Substitute \(x=\frac{184}{15}\) into \(3x - 5\), \(3\times\frac{184}{15}-5=\frac{552}{15}-5=\frac{552-75}{15}=\frac{477}{15}=31.8\) degrees.
For the angle corresponding to the 4 - ball (\(8x + 2\)):
Substitute \(x=\frac{184}{15}\) into \(8x + 2\), \(8\times\frac{184}{15}+2=\frac{1472}{15}+2=\frac{1472 + 30}{15}=\frac{1502}{15}\approx100.13\) degrees.

Step5: Determine the greatest - distance balls

In a triangle, the greatest side is opposite the greatest angle. The greatest angle is the one for the 4 - ball. So the greatest distance is between the balls corresponding to the 5 - ball and the 9 - ball.

Answer:

\(x=\frac{184}{15}\)
Measure of the angle for the 5 ball: \(\frac{721}{15}\) degrees
Measure of the angle for the 9 ball: \(\frac{477}{15}\) degrees
Measure of the angle for 4 ball: \(\frac{1502}{15}\) degrees
Between which two balls is the distance the greatest: 5 - ball and 9 - ball