Exploratory challenge
choose one of the colored pointsthat you marked. draw the right triangle formed by the line segment connecting the original two points and and that colored point.
take a copy of the triangle, and rotate it 180° about the midpoint of ab
label the acute angles in the original triangle as x and y, and label the corresponding angles in the rotated triangle the same.
todd says abcd′ is a rectangle. maryam says abcd′ is a quadrilateral, but she is not sure it is a rectangle. todd is right but does not know how to explain himself to maryam.
can you him out?
a. what composite figure is formed by the two triangles? how would you prove it?
i. what is the sum of the measures of x and y? why?
ii. how do we know that the figure whose vertices are the colored points …) and points and is a rectangle?
b. draw the two diagonals of the rectangle. where is the midpoint of the segment connecting the two original points and why?
c. label the intersection of the diagonals as point p. how does the distance from point p to a colored point …) compare to the distance from p to points and
d. choose another colored point, and construct a rectangle using the same process you followed before. draw the two diagonals of the new rectangle. how do the diagonals of the new and old rectangle compare? how do you know?
e. how does your drawing demonstrate that all the colored points you marked do indeed lie on a circle?