Let O be the the origin. OA means O to A and so on.
So if OD+OE+OF = 4(OA+OB+OC) , then △DEH is congurent to △ABC? Is this statement correct? If yes, why? How do we prove it? If no, how can i prove that the 2 triangles are congruent?
Thank you.
Let O be the the origin. OA means O to A and so on.
So if OD+OE+OF = 4(OA+OB+OC) , then △DEH is congurent to △ABC? Is this statement correct? If yes, why? How do we prove it? If no, how can i prove that the 2 triangles are congruent?
Thank you.
To be congruent the two triangles must have the same size and shape. If the lengths of the sides of triangle DEF add up to be 4 times longer than the lengths of the sides of triangle ABC they are not congruent. In addition, even if the lengths did add up to be the same you still wouldn't know whether the triangles are congruent without knowing whether the individual lengths are identical.
A couple of thoughts:
1. What is H?
2. On the face of it, I would definitely claim that congruency does not follow from the given equation. Assuming you meant triangle DEF congruent to triangle ABC, suppose you had OD = OE = OF = 4, and it was an equilateral triangle (that's not even required from those equations, but let's suppose that's the case). Now suppose OA = 0.5, OB = 1.5, and OC = 1. Then the equation holds, but you can see that congruency does not have to hold. Essentially, the problem is that the equations you are given do not place any constraint whatsoever on the angles of the triangle.
How bout if it is given OD=2OA+OB+OC, OE=OA+2OB+OC, and OF=OA+OB+2OC,such that ABC and DEF are two triangles in the plane and O is the origin? How can i prove that △ABC and △DEF are congruent?
Actually, in vector, what conditions make 2 triangles congruent?
The example you gave does indeed yield triangles ABC and DEF being congruent - it's not too difficult to show that length AB = length DE, length AC = length DF, and length BC = length DE. But the general case of (OD +OE+OF) = 4(OA+OB+OC) does not yield congruent triangles. For example if A = (1,0), B= (-1,0), and C=(1,1) you have (OA+OB+OC) = 1. then you could set D=(1,0), E=(-1,0), and F = (4,1), so that OD+OE+OF = 4. DEF meets the condition specified, but is not congruent to ABC.