Prove Triangle ABC Medians X,A,Y Lie in a Straight Line - Help Tanya

[Tanya Back]

Hey Guys! I am really stuck on this questions:

The medians BD and CE in triangle ABC are produced to X and Y respectively so that BD=DX and CE=EY. Prove that X,A, and Y lie in a straight line.

I tried this question, it made no sense to me...i drew the picture and i named the interesting point between BD and CE "F". Then i proved that Traingle BYF is congruent to triangle CXF (SAS)...and then i have noo clue wut to do..PLz Help!

Thank u!
Tanya

 

[Andrew Mason]

Hey Guys! I am really stuck on this questions:

The medians BD and CE in triangle ABC are produced to X and Y respectively so that BD=DX and CE=EY. Prove that X,A, and Y lie in a straight line.

Look at AXD and BCD and at AYE and BCE. Show that AX is parallel to BC and AY is parallel to BC (look at corresponding angles). If you can show this, then since AX and AY are parallel to the same line and share a common point, XAY would have to be a straight line.

You can also show that XAC = ACB and CBA = BAY. Since CAB is the third angle in triangle ABC, ACB + CBA + CAB = 180 = XAC + BAY + CAB. The last three angles sum to XAY which means that XAY is a straight line (180 degrees).

AM

 

[thepatientmental]

Hi Tanya,

Don't worry, proving that X, A, and Y lie in a straight line can be tricky at first but I will help you break it down step by step.

First, let's draw the triangle ABC with the medians BD and CE intersecting at point F, as you have done. We know that BD and CE are medians, which means they divide the sides of the triangle into two equal parts. This also means that point F is the centroid of the triangle ABC, which is the point of intersection of all three medians.

Now, let's focus on triangle BYF and CXF. As you mentioned, we can prove that they are congruent using the SAS (side-angle-side) congruence rule. This is because BY and CX are equal in length (since they are medians) and angle BYF is equal to angle CXF (since they are both right angles formed by the medians). Therefore, we can say that triangle BYF is congruent to triangle CXF.

Next, we can extend BY and CX to meet at point A. This creates two congruent triangles, triangle AYF and triangle AXF, since they share the side AF and have equal angles (angle AYF is equal to angle AXF because they are vertical angles).

Since triangle AYF and triangle AXF are congruent, this means that AY and AX are equal in length. And since we know that BY and CX are equal in length (since they are medians), this means that AY and AX are also equal to BY and CX respectively. Therefore, we can say that AYBY and AXCX are parallelograms.

Now, since AY and AX are parallelograms, this means that AY and AX are parallel to BY and CX respectively. And since we know that BY and CX intersect at point F, this means that AY and AX also intersect at point F. Therefore, we can conclude that X, A, and Y lie in a straight line, which is the line segment AF.

I hope this helps you understand and solve the problem. Keep practicing and you'll become a pro at proving geometric theorems in no time! Good luck!