Draw a triangle on tracing paper. Fold the altitude from each vertex.

Triangle is graphed.

1. Find the slope of each side of the triangle.
2. Find the slope of each altitude of the triangle.
3. Sketch the altitudes. Label the point of intersection, .
4. Write equations for all 3 altitudes.
5. Use the equations to find the coordinates of , and verify algebraically that the altitudes all intersect at .

Triangle is graphed.

1. Find the midpoint of each side of the triangle.
2. Sketch the perpendicular bisectors, using an index card to help draw 90-degree angles. Label the intersection point as .
3. Write equations for all 3 perpendicular bisectors.
4. Use the equations to find the coordinates of , and verify algebraically that the perpendicular bisectors all intersect at .

Consider triangle from an earlier activity.

1. What is the distance from to , the intersection point of the perpendicular bisectors of the triangle’s sides? Round to the nearest tenth.
2. Write the equation of a circle with center and radius .
3. Construct the circle. What do you notice?
4. Verify your hypothesis algebraically.

Consider triangle from earlier activities.

1. Plot point , the intersection point of the altitudes.
2. Plot point , the intersection point of the perpendicular bisectors.
3. Find the point where the 3 medians of the triangle intersect. Plot this point and label it .
4. What seems to be true about points and ? Prove that your observation is true.

A tessellation covers the entire plane with shapes that do not overlap or leave gaps.

1. Tile the plane with congruent rectangles:
   1. Draw the rectangles on your grid.
   2. Write the equations for lines that outline 1 rectangle.
2. Tile the plane with congruent right triangles:
   1. Draw the right triangles on your grid.
   2. Write the equations for lines that outline 1 right triangle.
3. Tile the plane with any other shapes:
   1. Draw the shapes on your grid.
   2. Write the equations for lines that outline 1 of the shapes.