1. D is a point on the side AC of a triangle ABC. Is ABC isoceles?
1. The area of triangular region ABD = area of BDC.
2. BD perpendicular AC and AD = DC.
Try draw a graph yourself and you'll know that this will be true if AD=DC, even if when ABC is not isoceles. If BD is perpendicular to AC then ABC is isoceles, otherwise it is not.
Triangle ABD and triangle CBD are similar (AD=CD, BD=BD, angle BDA=angle BDC), therefore BA=BC.
2. If the area of triangular region ABC is 25, what is the perimeter of ABC?
1. The length of one side of ABC is 5sqrt(2).
2. ABC is a right isoceles triangle.
Say angle ABC is 90 degree, then we know that the other two angles are 45 degrees and the three sides are 1:1:sqrt(2). From S=1/2BA*BC=25 and BA=BC we can solve for BA, then the other two sides, and thus the perimeter.
Keep on asking, and it will be given you;
keep on seeking, and you will find;
keep on knocking, and it will be opened to you.