Circle-Inscribed Quadrilateral and Similar Triangles
Circle-Inscribed Quadrilateral and Similar Triangles
1) Drag the points of the quadrilateral around the circle (do not cross adjacent points (yet)).
2) Observe the sum of opposite angles as the points move around the circle (still no crossing).
3) Observe the sum of adjacent angles as the points move around the circle (still no crossing).
4) Use the slider to dilate the circle. Notice anything changing (other than the radius)? No
5) Now drag one of the points across an adjacent point creating an intersection and two small triangles.
6) Now two of the points have an exterior angle measure and two of the points have an interior angle measure.
7) What do you notice about the same colored angles' sums? (hint: look at the purple and green sums)
8) What does this suggest about the interior angles of the same color?
9) Given that each of the two smaller triangles have two angles congruent (purple and green), what does that imply about the third angle of the two triangles (at the intersection)?
10) What does that imply about the two triangles you've created? (think AAA)
11) Adjust the slider to 1 or 2 (maybe 3 if using a tablet or small screen). Hold a thin piece of paper over the screen and trace the larger of the two triangles with a pen or pencil. Now flip the piece of paper over, place the triangle you traced over the smaller triangle on the screen and begin adjusting the slider up until the smaller triangle fits the triangle you traced (you may need to rotate the paper). You can also move the center of the circle if you need more room to work.
What is the sum of opposite angles of a circle-inscribed quadrilateral?
What is the sum of adjacent angles in a circle-inscribed quadrilateral?
After crossing the points to form two triangles, what do you notice about the interior measures of same colored angles?
What can we say about the two triangles created in the circle?