MPM1D Midpoints and Medians in Triangles

Midpoint: B A

C

1) Use a ruler to create a midpoint for AB. Label it X.

2) Use a ruler to create a midpoint for AC. Label it Y.

3) Join the points X and Y with a line.

4) Measure all the sides. Write them on the diagram.

Q: What do you notice about the length of XY and BC?

Q: What do you notice about the direction of the lines XY and BC?

Q: How could we prove this?

The line segments joining the midpoint of two sides of a triangle is ______the third side and ______.

Recall: The area of a triangle is: � =

Median: A line from one vertex to the midpoint of the opposite side.

A

B C

1) Draw the midpoint of BC, label it M 2) Draw the median from A to BC 3) Calculate the area of rABM and rAMC (Measuring any lengths you need to find the area)

Medians of a triangle ______its area.

Example 2 The area of rXYZ is 48 ��. A median is drawn from X, and the midpoint of the opposite side is labelled M. What is the area of rXYM?

1. Find the length of line segment MN in each triangle. a) b) c) d)

2. Find the lengths of line segments AD and DE in each triangle. a) b)

c) d)

3. The area of rABC is 10 cm2. Calculate the area of each triangle. a) rABD b) rADC

4. Calculate the area of each triangle given the area of rMNQ is 12 cm2. a) rMNP b) rNPQ

5. Here are two conjectures about a median in an isosceles triangle. For each conjecture, explain why the conjecture is true, or draw a counterexample to show it is false. a) The median to the vertex opposite the unequal side bisects the angle. b) The median to a vertex opposite one of the equal sides bisects the angle.

Answer

1. a) 31 m b) 1.5 cm c) 54 cm d) 4 m 2. a) AD = 15 m; DE = 20 m b) AD = 4 cm; DE = 5.8 cm c) AD = 60 m; DE = 104 m d) AD = 6 cm; DE = 13.2 cm 3. a) 5 cm2 b) 5 cm2 4. a) 24 cm2 b) 12 cm2 5. a) The median divides the isosceles triangle into two smaller triangles. It bisects the unequal side, and is a common side in each of the smaller triangles. The original triangle is isosceles, so the third sides in the smaller triangles are equal. The angles opposite the new common side were equal and still are, and both new triangles have a right angle where the common side meets the original unequal side. Therefore, the two triangles have all sides equal and two angles equal, so the third angle in each must be equal. So, the median to the vertex opposite the unequal side of an isosceles triangle bisects the angle at that vertex. b) False