Angles and Secants P2
Drag the green dots around to see the theorems and equations at work.
THEOREM: For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle.
(3)
A) Place the point P inside of the circle. What is the product of the two pieces of the pink line segment? What is the product of the two pieces of blue segment?
THEOREM: For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and the circle.
(3)
B) Place the point P outside of the circle. What is the distance from P to the green point on the pink line? From P to the blue point on the pink line?. What is the product of those two lengths? What is the distance from P to the green point on the blue line? From P to the blue point on the blue line?. What is the product of those two lengths?
C) Place the point P outside of the circle so the pink segment does not go into the circle (the green dot matches with the blue dot).
i) What is the distance from P to the green point on the blue line? From P to the blue point on the blue line?. What is the product of those two lengths?
ii) What is the distance from P to the green point on the pink line? From P to the blue point on the pink line?. What is the product of those two lengths?
D) Did you expect these relationships? Why or why not?
E) State the theorem in your own words.
THEOREM: The measure of an angle formed by two lines that intersect outside a circle is half the difference of the measures of the intercepted arcs.
(4)
A) What is the difference between the measure of Arc 1 and Arc 2? What is half of that value?
B) How does that number compare to the value of angle P? Does that surprise you? Why or why not?
C) State this theorem in your own words.