# Thread: Geometry Properties of a Triangle Help

1. ## Properties of a Triangle

I need help on using midpoints to draw a triangle... here's the question

"the midpoints of the sides of a triangle are L(4,2), M(2,3), AND N(5,4). What are the coordinates of the vertices of the triangle?"

I understand that you find the slope of each side, but what do you do after that? Please help and thanx

2. Originally Posted by I'm_Schuyler
I need help on using midpoints to draw a triangle... here's the question

"the midpoints of the sides of a triangle are L(4,2), M(2,3), AND N(5,4). What are the coordinates of the vertices of the triangle?"

I understand that you find the slope of each side, but what do you do after that? Please help and thanx
1. Let A, B and C denote the vertices of the triangle you are looking for.

2. Draw the triangle LMN.

3. The side of triangle ABC which passes through L must be parallel to MN:

Calculating the slope MN: $\displaystyle m_{MN}=\dfrac{3-4}{2-5}=\dfrac13$
Calculating the equation of the line through L:
$\displaystyle y-2=\frac13 (x-4)~\implies~\boxed{y = \frac13x + \frac23}$

4. Do just the same with the points M and N. You then have 3 equations of three lines.

5. Calculate the coordinates of the intersection points of these three lines.

3. 3. The side of triangle ABC which passes through L must be parallel to MN:

I dont get how the line passing through L is parallel to MN.
Any explanation?

4. Originally Posted by syathish
3. The side of triangle ABC which passes through L must be parallel to MN:

I dont get how the line passing through L is parallel to MN.
Any explanation?
There is a theorem whos name is (literally translated) "mid parallel in triangles theorem ":

"The connection between 2 midpoints of 2 sides in a triangle is parallel to the third side and has half the length of the third side."

I've used this theorem.

5. Originally Posted by earboth
There is a theorem whos name is (literally translated) "mid parallel in triangles theorem ":

"The connection between 2 midpoints of 2 sides in a triangle is parallel to the third side and has half the length of the third side."

I've used this theorem.
Thnx fo reminding me m8