Question

solving real - world problems with trapezoids
a plot of land has been surveyed for a new housing
development with borders ab, bc, dc, and da. the plot
of land is a right trapezoid with a height of 60 feet and an
opposite leg length of 65 feet.
if the measure of angle bcd is 67.4°, the measure of
angle abc is
if the length of base ab is 80 feet, the length of dc is
feet.

Explanation:

Step1: Find the measure of angle \(ABC\)

Since \(AB\parallel DC\) in trapezoid \(ABCD\), \(\angle ABC+\angle BCD = 180^{\circ}\) (co - interior angles).
Given \(\angle BCD=67.4^{\circ}\), then \(\angle ABC=180^{\circ}-\angle BCD\)
\(\angle ABC = 180 - 67.4=112.6^{\circ}\)

Step2: Find the length of \(DC\)

Let's draw a perpendicular from \(B\) to \(DC\) and call the foot of the perpendicular \(E\).
We know that \(AD = BE=60\) feet (height of the trapezoid) and \(BC = 65\) feet.
Using the Pythagorean theorem in right - triangle \(BEC\), we can find \(EC\). Let \(EC=x\), then \(x=\sqrt{BC^{2}-BE^{2}}\) (by \(a^{2}+b^{2}=c^{2}\), where \(c = BC\), \(a = BE\), \(b = EC\))
\(x=\sqrt{65^{2}-60^{2}}=\sqrt{(65 + 60)(65 - 60)}=\sqrt{125\times5}=\sqrt{625}=25\)
Since \(AB = DE = 80\) feet (because \(ABED\) is a rectangle), and \(DC=DE + EC\)
\(DC=80 + 25=105\) feet

Answer:

The measure of angle \(ABC\) is \(112.6^{\circ}\). The length of \(DC\) is \(105\) feet.