SOLUTION: In triangle ABC, D is a point on segment AC and E is a point on CB such that DE and AB are parallel segments. If AD and DC are congruent and DE= 3, find the length of AB

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Question 1070409: In triangle ABC, D is a point on segment AC and E is a point on CB such that DE and AB are parallel segments. If AD and DC are congruent and DE= 3, find the length of AB
Found 3 solutions by KMST, solver91311, ikleyn:
Answer by KMST(5328) About Me  (Show Source):
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DE is obviously the midsegment of ABC that is parallel to AB,
because AD=DC makes D the midpoint of AC.
We immediately think that DE=%281%2F2%29+AB , so highlight%28AB=6%29 ,
but we have not proven that.
As proof, we could offer that triangles ABC and DEC are similar,
because they share angle ACB,
and have another pair of congruent angles in
corresponding angles CAB and CDE,
formed by parallel lines AB and BC with transversal AC.
Since AD=CD=DC, AC=AD+DC=2DC,
so the length rato of corresponding sides is
AC%2FDC=2
and applied to another pair of corresponding sides,
AB%2FDE=2--->AB=2E-->AB=2%2A3--->highlight%28AB=6%29%0D%0A .

The problem is that the midsegment theorem starts with the hypothesis that
H: the ends of a segment are the midpoint of 2 sides of a triangle,
and concludes (thesis) that
T: the segment is parallel to the third side and half as long.
We need another theorem.
It would say that if
H: a line goes through the midpoint of a side of a triangle,
and is parallel to another (second side),
then
T: the line intercepts the third side at its midpoint,
and the segment of that line inside the triangle is half as long as the parallel side.
We cannot even call that new theorem the midsegment converse,
because we did not simply swap H and T,
Each theorem has a couple of facts for H and another couple for T,
but the couples have rearranged themselves, changing partners.

Answer by solver91311(24713) About Me  (Show Source):
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AB parallel to DE so CA is a transversal of two parallel lines. Hence angle CDE congruent to angle CAB and angle CED congruent to angle CBA. Angle C is congruent to itself. So by AAA, triangle DEC is similar to triangle ABC.

Since AD is congruent to DC, D must be the midpoint of AC. Therefore AC is twice the measure of AD or DC. Therefore the sides of triangle DEC and the sides of triangle ABC are in proportion 1:2.

John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52756) About Me  (Show Source):
You can put this solution on YOUR website!
.
Theorem 1
The straight line connecting midpoints of two sides of a triangle is parallel to the third side of the triangle.


Theorem 2
The straight segment connecting midpoints of the two sides of a triangle is of half of the length of the third side of the triangle.


Theorem 3
If a straight line bisects one side of a triangle and is parallel to a second side of the triangle, then it bisects the third side of the triangle.


All these theorems are proved in the lesson
    - The line segment joining the midpoints of two sides of a triangle
in this site.

Also,  you have this free of charge online textbook on Geometry
    GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.

The referred lesson is the part if this textbook under the topic "Properties of triangles".


The textbook contains EVERYTHING that you ever need to know/to use of Geometry in very organized and structured form.


          H A P P Y    L E A R N I N G  ! !