Paper Folding The figures below show how to construct altitudes and medians by paper folding. Refer to them for Exercises 29 and 30.

Folding an Altitude

Fold the triangle so that a side A C ¯ eh c bar  overlaps itself and the fold contains the opposite vertex B.

Folding a Median

Fold one vertex R to another vertex P. This locates the midpoint M of a side.

Unfold the triangle. Then fold it so that the fold contains the midpoint M and the opposite vertex Q.

29. Cut out a large triangle. Fold the paper carefully to construct the three medians of the triangle and demonstrate the Concurrency of Medians Theorem. Use a ruler to measure the length of each median and the distance of each vertex from the centroid.
30. Cut out a large acute triangle. Fold the paper carefully to construct the three altitudes of the triangle and demonstrate the Concurrency of Altitudes Theorem.

31. In the figure below, C is the centroid of Δ D E F . cap delta d e f .  If G F = 12 x 2 + 6 y , g f equals . 12 , x squared . plus 6 y , comma  which expression represents CF?

    

    1. 6 x 2 + 3 y 6 , x squared . plus 3 y
    2. 4 x 2 + 2 y 4 , x squared . plus 2 y
    3. 8 x 2 + 4 y 8 , x squared . plus 4 y
    4. 8 x 2 + 3 y 8 , x squared . plus 3 y
32. Reasoning What type of triangle has its orthocenter on the exterior of the triangle? Draw a sketch to support your answer.
33. Writing Explain why the median to the base of an isosceles triangle is also an altitude.
34. Coordinate Geometry Δ A B C cap delta eh b c  has vertices A(0, 0), B(2, 6), and C(8, 0). Complete the following steps to verify the Concurrency of Medians Theorem for Δ A B C . cap delta eh b c .

    

    1. Find the coordinates of midpoints L, M, and N.
    2. Find equations of A M ↔ , B N ↔ , modified eh m with left right arrow above , comma . modified b n with left right arrow above , comma  and C L ↔ . modified c l with left right arrow above , .
    3. Find the coordinates of P, the intersection of A M ↔ modified eh m with left right arrow above  and B N ↔ . modified b n with left right arrow above , .  This point is the centroid.
    4. Show that point P is on C L ↔ . modified c l with left right arrow above , .
    5. Use the Distance Formula to show that point P is two-thirds of the distance from each vertex to the midpoint of the opposite side.

C Challenge

35. Constructions A, B, and O are three noncollinear points. Construct point C such that O is the orthocenter of Δ A B C . cap delta eh b c .  Describe your method.
36. Reasoning In an isosceles triangle, show that the circumcenter, incenter, centroid, and orthocenter can be four different points, but all four must be collinear.

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