Download Explore Ratios, Proportions, and Equalities within a Triangle

Explore Ratios, Proportions, and Equalities within a Triangle
Directions:
1. Determine which statements are true.
2. Provide a counterexample for the statements that are false.
3. Conjecture and explore on your own to find additional true statements,
or modify a statement to make it true.
4. Prove the statements that are true.
Conjecture 1: In any triangle, the ratio of any two sides is equal to the ratio of the
corresponding altitudes.
Conjecture 2: If a line parallel to one side of a triangle intersects the other two sides,
then it divides these sides into segments that are proportional.
Conjecture 3: In any triangle, the bisector of any angle of the triangle divides the
opposite side into segments proportional to the other two sides.
Conjecture 4: In any triangle, the segment whose endpoints are the midpoints of two
sides of the triangle is half the length of the third side.
Conjecture 5: Given a value r between 0 and 1. In any triangle, the segment whose
endpoints are r times the length of the respective side from a given vertex has length r
times the length of the third side.
Conjecture 6: In any triangle, the ratio of any two medians is equal to the ratio of the
corresponding altitudes.
Conjecture 7: Any two medians of a triangle intersect at a point that is two-thirds of the
distance from any vertex to the midpoint of the opposite side.
Conjecture 8: Given ABC. Let D and E be the foot of the altitudes to vertices A and B,
respectively. Let F be the point of intersection of the lines containing the two altitudes.
AF FE

.
Then
BF FD
Conjecture 9: Any two angle bisectors of a triangle intersect at a point equal distant from
all three sides of the triangle.
Conjecture 10: Any two perpendicular bisectors of two sides of a triangle intersect at a
point equal distant from all three vertices of the triangle.
Solutions
Conjecture 1:
Conjecture 4:
Conjecture 6:
Conjecture 8:
False, modify to make true. Conjecture 2: True.
Conjecture 3: False, modify to make true.
True.
Conjecture 5: True. Similar to Conjecture 4 use SAS Similarity.
False.
Conjecture 7: True
True. Prove by using AA Similarity with several cases:
(1) If the intersection point for the altitudes is internal to the triangle use vertical angles and right angles.
(2) If the intersection point for the lines containing the altitudes is external to the triangle, use the common angle at
the intersection point and right angles for the overlapping triangles.
Conjecture 9: True.
Conjecture 10: True.