Answer:
- diagonal: 20.68 feet
- short base: 17.21 feet
Step-by-step explanation:
You want the length of the diagonal and the short base of an isosceles trapezoid with long base 22 feet and sides 7 feet. The base angles total 140°.
Diagonal
The base angles are congruent, so each of them is 140°/2 = 70°. The length of the diagonal can be found from the law of cosines (reference the attached figure).
[tex]AC^2=AD^2+CD^2-2AD\cdot CD\cdot\cos(D)\\\\AC^2=22^2+7^2-2\cdot22\cdot7\cdot\cos(70^\circ)\approx427.6578\\\\AC\approx\sqrt{427.6578}\approx20.68[/tex]
The length of the diagonal is about 20.68 feet.
Short Base
We can find the measure of angle C in ∆ABC using the law of sines.
[tex]\dfrac{\sin(C)}{AB}=\dfrac{\sin(B)}{AC}\\\\\\C=\sin^{-1}\left(\dfrac{AB}{AC}\sin(B)\right)=\sin^{-1}\left(\dfrac{7}{20.68}\sin(110^\circ)\right)\\\\C\approx18.55^\circ[/tex]
Angle A brings the total of angles in ∆ABC to 180°:
A = 180° -110° -18.55° = 51.45°
Using the law of sines again, we find the short base to be ...
[tex]\dfrac{BC}{\sin(A)}=\dfrac{AB}{\sin(C)}\\\\\\BC=AB\cdot\dfrac{\sin(A)}{\sin(C)}=7\cdot\dfrac{\sin(51.45^\circ)}{\sin(18.55^\circ)}\\\\BC\approx17.21[/tex]
The length of the short base is about 17.21 feet.
