91Ó°ÊÓ

Consider two chords that intersect at point \(P\) .Do you think that \(\frac{AP}{BP}=\frac{CP}{DP} ?\) Justify your answer.

Circumscribe a triangle about a circle. Then, using the points of tangency, inscribe a triangle in the circle. Must it be true that the two triangles are similar? Explain your reasoning.

The vertices of \(\Delta\) XY Zare \(X(4,5),\) \(Y(4,13),\) and \(Z(8,9)\) . Find the equation of the circle circumscribed about \(\Delta X Y Z\) Justify your answer.

Find the missing interior angle measure. Pentagon \(PQRST\) has angle measures \(m \angle P=85^{\circ}, m \angle Q=134^{\circ}, m \angle R=97^{\circ},\) and \(m \angle S=102^{\circ} .\) Find \(m \angle T .\)

A circle has center (h, k) and contains point (a, b). Write the equation of the line tangent to the circle at point (a, b).

MAKING AN ARGUMENT Your friend claims that it is possible for a circumscribed angle to have the same measure as its intercepted arc. Is your friend correct? Explain your reasoning.

REASONING Points A and B are on a circle, and t is a tangent line containing \(\mathrm{A}\) and another point \(\mathrm{C}\) . a. Draw two diagrams that illustrate this situation. b. Write an equation for mAB in terms of \(\mathrm{m} \angle \mathrm{BAC}\) for each diagram. c. For what measure of \(\angle \mathrm{BACcan~you~use~either~}\) equation to find mAB? Explain.

MODELING WITH MATHEMATICS Three moons, A, B, and C, are in the same circular orbit 100,000 kilometers above the surface of a planet. The planet is 20,000 kilometers in diameter and \(\mathrm{m} \angle \mathrm{ABC}=90^{\circ} .\) Draw a diagram of the situation. How far is moon A from moon \(\mathrm{C}\) ?

WRITING A right triangle is inscribed in a circle, and the radius of the circle is given Explain how to find the length of the hypotenuse.

For a point outside of a circle, how many lines exist tangent to the circle that pass through the point? How many such lines exist for \(\square\) a point on the circle? inside the circle? Explain your reasoning.