SOLUTION: given:triangle abc, ab=ac, d is the point between b and c. prove: ab>ad so far i wrote the 2 givens and the third statement which is <c=<d because if 2 ext sides are congruent

Algebra ->  Geometry-proofs -> SOLUTION: given:triangle abc, ab=ac, d is the point between b and c. prove: ab>ad so far i wrote the 2 givens and the third statement which is <c=<d because if 2 ext sides are congruent      Log On


   

Question 826466: given:triangle abc, ab=ac, d is the point between b and c.
prove: ab>ad
so far i wrote the 2 givens and the third statement which is AD because of the substitution, and longer sides are across from longer angles. Now, i need to write from statement 4 to 7 explaining how i got to my last 8th prove but i don't know how and what to write. Help please.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
There are two possibilities:
  • Segment AD is perpendicular to to segment BC. If this is true, then triangle ADB is a right triangle (since perpendiculars form right angles). And AD is a leg of this triangle and AB is the hypotenuse. Hypotenuse's are always the longest side of a right triangle (since it is opposite the largest angle).
  • It is unknown if AD is perpendicular to BC. In this case:
    1. Draw a perpendicular from A to BC and label the point on BC as point E.
    2. We now have two right triangles: AEB and AED.
    3. Both right triangles share the leg AE.
      4. The other legs of these triangles are EB for triangle AEB and ED for triangle AED.
    4. EB > ED
    5. AB is the hypotenuse of triangle AEB and AD is the hypotenuse of triangle AED.
    6. Triangle AEB, with one leg identical and the other leg longer than triangle AED's, must have a longer hypotenuse.
    7. So AB > AD.
Note: If point D is closer to point C than to point B then the logic above will not be as clear. But if you change the references from AB to AC then the logic still works. And, if AC > AD and AB = AC, then AB > AD.