Question

1. give the name for a regular polygon in which each exterior angle measures 18°. 24 - gon 22 - gon 18 - gon 15 - gon none of these answers are correct. 8. given that the quadrilateral mnop is a trapezoid, \\(\overline{qr}\\) is the median, \\(\overline{qr}=27\\) and \\(\overline{op}=37\\), what is the length of \\(\overline{mn}?\\)

Explanation:

Step1: Recall exterior - angle formula for regular polygons

The sum of exterior angles of any polygon is \(360^{\circ}\). For a regular polygon with \(n\) sides, each exterior angle \(\theta=\frac{360^{\circ}}{n}\).

Step2: Solve for \(n\)

Given \(\theta = 18^{\circ}\), we set up the equation \(18^{\circ}=\frac{360^{\circ}}{n}\). Cross - multiply to get \(18n = 360\). Then \(n=\frac{360}{18}=20\). Since none of the options match \(20 -\)gon, the answer to question 7 is "None of these answers are correct."

Step3: Recall trapezoid median formula

The formula for the length of the median of a trapezoid is \(QR=\frac{MN + OP}{2}\), where \(QR\) is the median, \(MN\) is one base, and \(OP\) is the other base.

Step4: Solve for \(MN\)

We know \(QR = 27\) and \(OP=37\). Substitute into the formula \(27=\frac{MN + 37}{2}\). Multiply both sides by 2: \(27\times2=MN + 37\), so \(54=MN + 37\). Then \(MN=54 - 37=17\).

Answer:

1. None of these answers are correct.
2. \(17\)