91Ó°ÊÓ

Discover and prove a theorem about two lines tangent to a circle at the endpoints of a diameter.

The latitude of a city is given. Sketch the Earth and a circle of latitude through the city. Find the radius of this circle. Milwaukee, Wisconsin; \(43^{\circ} \mathrm{N}\)

Draw two congruent circles with radii 6 each passing through the center of the other. Find the length of their common chord.

A plane \(P\) cuts sphere \(O\) in a circle that has diameter \(20 .\) If the diameter of the sphere is \(30,\) how far is the plane from \(O ?\)

Use trigonometry to find the measure of the arc cut off by a chord \(12 \mathrm{cm}\) long in a circle of radius \(10 \mathrm{cm} .\)

\(\overline{P X}\) and \(\overline{P Y}\) are tangents. If \(m \angle P=65,\) then \(m \widehat{X Y}=\underline{?}\)

A line is tangent to two intersecting circles at \(P\) and \(Q .\) The common chord is extended to meet \(\overline{P Q}\) at \(T .\) Prove that \(T\) is the midpoint of \(\overline{P Q}\).

Investigate the possibility, given a circle, of drawing two chords whose lengths are in the ratio 1: 2 and whose distances from the center are in the ratio \(2: 1 .\) If the chords can be drawn, find the length of each in terms of the radius. If not, prove that the figure is impossible.

Equilateral \(\triangle A B C\) is inscribed in a circle. \(P\) is any point on \(\widehat{B C}\). Prove \(P A=P B+P C .\) (Hint: Use Ptolemy's Theorem.)

Angle \(C\) of \(\triangle A B C\) is a right angle. The sides of the triangle have the lengths shown. The smallest circle (not shown) through \(C\) that is tangent to \(\overline{A B}\) intersects \(\overline{A C}\) at \(J\) and \(\overline{B C}\) at \(K\). Express the distance \(J K\) in terms of \(a . b .\) and \(c .\)