Question

1. use the relationships in the diagrams below to write an equation and solve for x.

a. x 30 18
b. 2x x 2x + 20° 3x + 20°

Explanation:

Step1: Apply Pythagorean theorem for part a

In a right - triangle, by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. Here, \(x^{2}+18^{2}=30^{2}\).
\[x^{2}+324 = 900\]

Step2: Solve for \(x^{2}\) in part a

Subtract 324 from both sides of the equation:
\[x^{2}=900 - 324=576\]

Step3: Solve for \(x\) in part a

Take the square root of both sides. Since \(x\) represents a length, we take the positive square root.
\[x=\sqrt{576}=24\]

Step4: Use angle - sum property for part b

The sum of angles around a point is \(360^{\circ}\). So, \(x + 2x+(2x + 20^{\circ})+(3x + 20^{\circ})=360^{\circ}\).
\[x+2x + 2x+3x+20^{\circ}+20^{\circ}=360^{\circ}\]
\[8x+40^{\circ}=360^{\circ}\]

Step5: Solve for \(x\) in part b

Subtract \(40^{\circ}\) from both sides: \(8x=360^{\circ}- 40^{\circ}=320^{\circ}\).
Then divide both sides by 8: \(x = 40^{\circ}\)

Answer:

a. \(x = 24\)
b. \(x = 40^{\circ}\)