Medians AX and BY of Triangle ABC are perpendicular at point G. Prove that AB=CG.
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To prove that AB = CG, we can use the properties of medians and the given information that medians AX and BY are perpendicular at point G in triangle ABC.
Let's break down the steps to prove this:
Step 1: Draw the triangle ABC and label the medians AX and BY.
Step 2: Since AX and BY are medians, they divide each other into segments of the same length. Therefore, label the intersection point of AX and BY as point G.
Step 3: We know that medians AX and BY intersect at right angles at point G. This implies that AG and BG, the segments connecting vertices A and B to the midpoint of the opposite sides, are equal in length.
Step 4: Next, we need to prove that CG is also equal to AB. To do this, we will prove that triangle ACG and triangle BCG are congruent.
Step 5: We already know that AG = BG (as we proved in step 3). Now, we need to prove that angle ACG = angle BCG.
Step 6: Since AX and BY are medians, they divide their respective opposite sides (BC and AC) into two equal halves. Therefore, the triangles ACG and BCG share the base CG and the height AG = BG.
Step 7: Since the base and height of both triangles are equal, the triangles ACG and BCG are congruent by the Side-Angle-Side (SAS) congruence criterion.
Step 8: When two triangles are congruent, their corresponding sides are equal. In this case, since triangle ACG and triangle BCG are congruent, we can conclude that AC = BC.
Step 9: Finally, using the transitive property of equality, we have AC = BC and AG = BG. Therefore, AB = CG.
Therefore, we have proved that AB = CG using the properties of medians and the given information that medians AX and BY are perpendicular at point G in triangle ABC.