Please help with this problem!?

Question

Prove that the line segment joining the midpoints of the two sides of a triangle is parallel to the third side.
If it helps, use the points (0,0) (8,0) (2,5)

Answer 1

Get the midpoints for 2 sides. Get the slope of the 2 midpoints.

Get the slope of the 3rd side.

If the slopes are the same, then parallel.

Answer 2

There are probably many ways to prove this, but one I can think of is prove that the angle between the two lines is 0 degrees, or equivalently that the angle with the x-axis is the same.

Midpoints are (4,0) on the x-axis and (5,2.5). The latter you can work out using a triangle within the triangle on points (2,5) (2,0) and (8,0). (the midpoint of the hypotenuse of a right angle triangle will be found where lines perpendicular to the midpoints of the two other sides intersect. Midpoint along the x-axis is 3 units along from 2, so 5. Midpoint along the y direction is 5/2 = 2.5)

The angle made by the 3rd side with the x-axis is:
tan(A1) = 5/2 -> A1 = 68.19 degrees

Angle made with the midpoint line and the x-axis is:
tan(A2) = 2.5 / 1 -> A2 = 68.19 degrees

therefore the lines must be parallel.

There is probably another easier way to do this, but this one popped into my head right away.

Answer 3

Nobody remembers high-school geometry??

The midpoints of the sides means that the length of the new sides is exactly 1/2 that of the original sides and, since the included angle is the same, the new triangle (formed by 1/2 of the original 2 sides and the segment joining the midpoints) is 'similar' to the original triangle. In particular, all of the angles in the new triangle are the same as the angles in the original triangle and the proof is complete.

Q.E.D.


Doug

Answer 4

Several ways that this can be done.

First, you can find the line segment that connects the midpoints. Then find the angles of that line segment to the sides of the triangle. By the law of exterior angles, they are the same so therefore your line segment is parrallel to the side of the triangle. The exterior law states that the angles can only be equivalent if the 2 lines are parallel.

Secondly, you can find the slopes of the adjoining line segment and that of the third side.

Thirdly, you can prove that the new segment forms an interior triangle which is similar to the original one. By similar I mean they only differ by multiplication of a constant (shrink or expand). Such transformation do not affect slope.

Answer 5

Let triangle ABC have midpoints O at AB and P at BC which forms a similar triangle OBP since AB:OB = CB:PB and both triangles share a common vertex angle at ABC. Since the two triangles are similar then angle BOP = angle BAC and angle BPO = angle BCA, thus line segments OP and AC are parallel due to the transversal line rule (see http://en.wikipedia.org/wiki/Transversal_Line )

Answer 6

do not use numerical values to prove a theorem.

In the simplest form take the vertices as A(0,0), B(a,0) and C(b,c)

Consider midpoints of AB and BC

midpoint of AB is P(a/2,0)
midpoint of BC is Q((b+a)/2 , c/2)

slope of PQ is {c/2 -0} / {(b+a)/2 - a/2} = c/b

slope of AC is {c-0}/{b-0} = c/b

therefore slope of PQ = slope of AC

therefore PQ parallel to AC.

similarly prove for the other two sides