Question

find the missing length.
your answer
solve for x.

Explanation:

First Sub - Question: Find the missing length

Step1: Identify the trapezoid property

In a trapezoid, if a line segment is drawn parallel to the two bases, it divides the non - parallel sides proportionally. Also, the length of the mid - segment (the segment parallel to the bases) is the average of the lengths of the two bases. Here, we can assume that \(XW\) is the mid - segment of trapezoid \(GDEF\) (since \(XW\) is parallel to \(GD\) and \(FE\)). The formula for the mid - segment \(m\) of a trapezoid with bases \(b_1\) and \(b_2\) is \(m=\frac{b_1 + b_2}{2}\). Here, \(b_1 = 11.4\) and \(b_2=4.2\).

Step2: Calculate the mid - segment length

Substitute the values of \(b_1\) and \(b_2\) into the formula: \(m=\frac{11.4 + 4.2}{2}=\frac{15.6}{2}=7.8\)

Step1: Recall the trapezoid mid - segment formula

For trapezoid \(UVWT\), the mid - segment \(MN\) has length given by the formula \(MN=\frac{UV + TW}{2}\). We know that \(UV = 15\), \(TW = 21\) and \(MN=5x - 27\).

Step2: Substitute the values into the formula

Substitute into the formula: \(5x-27=\frac{15 + 21}{2}\)

Step3: Simplify the right - hand side

First, calculate the right - hand side: \(\frac{15 + 21}{2}=\frac{36}{2}=18\)

Step4: Solve for \(x\)

We have the equation \(5x-27 = 18\). Add 27 to both sides: \(5x=18 + 27=45\). Then divide both sides by 5: \(x=\frac{45}{5}=9\)

Answer:

\(7.8\)

Second Sub - Question: Solve for \(x\)