91Ó°ÊÓ

Find the radii \(R\) and \(r\) of the circumscribed and inscribed circles, respectively, of the triangle \(\triangle A B C\). \(a=5, b=7, C=40^{\circ}\)

Use the Law of Tangents to solve the triangle \(\triangle A B C\). \(a=7, B=60^{\circ}, c=9\)

Draw the triangle \(\triangle A B C\) and its circumscribed and inscribed circles accurately, using a ruler and compass (or computer software). \(a=2\) in, \(b=4\) in, \(c=5\) in

Two trains leave the same train station at the same time, moving along straight tracks that form a \(35^{\circ}\) angle. If one train travels at an average speed of \(100 \mathrm{mi} / \mathrm{hr}\) and the other at an average speed of \(90 \mathrm{mi} / \mathrm{hr}\), how far apart are the trains after half an hour?

Draw a circle with a radius of 2 inches and inscribe a triangle inside the circle. Use a ruler and a protractor to measure the sides \(a, b, c\) and the angles \(A, B, C\) of the triangle. The Law of Sines says that the ratios \(\frac{a}{\sin A}, \frac{b}{\sin B}, \frac{c}{\sin C}\) are equal. Verify this for your triangle. What relation does that common ratio have to the diameter of your circle?

For any triangle \(\triangle A B C\), show that \(c=b \cos A+a \cos B\). This is another check of a triangle.

Show that for any triangle \(\triangle A B C\), the radius \(R\) of its circumscribed circle is $$R=\frac{a b c}{\sqrt{(a+b+c)(b+c-a)(a-b+c)(a+b-c)}} .$$

An observer on the ground measures an angle of inclination of \(30^{\circ}\) to an approaching airplane, and 10 seconds later measures an angle of inclination of \(55^{\circ}\). If the airplane is flying at a constant speed and at a steady altitude of \(6000 \mathrm{ft}\) in a straight line directly over the observer, find the speed of the airplane in miles per hour. (Note: 1 mile \(=5280 \mathrm{ft}\) )

Show that for any triangle \(\triangle A B C\), the radius \(R\) of its circumscribed circle and the radius \(r\) of its inscribed circle satisfy the relation $$r R=\frac{a b c}{2(a+b+c)}$$