can solve?if in a triangle heigh(altitude)line=median(line)that triangle=isosceles?

Question

i solved this problem with pictucre of triangle i tried to send by attach this shape that i draw exactly but i couldnot success up to now please advise me i draw triangle and all formula thankfrei

Answer 1

i think what you're asking is whether a triangle whose median coincides with its altitude is necessarily isosceles. The answer is yes.

The altitude is by definition perpindicular to the base, and the median divides the base into two equal segments. If you draw this, you will see that the median/altitude divides the triangle into two congruent subtriangles (via SAS congruency). Thus the hypotenuses of these subtriangles are equal, and hence the triangle is isosceles.

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Answer 2

If in a triangle the median and the altitude coincide, the triangle is isoceles. Drop a perpendicular to the base and form two right triangles. If the altitude is also a median, it bisects the base and the two right triangles are congruent SAS=SAS. The sides of the original triange (the hypotenuses of the right triangles) are then congruent because they are corresponding parts of congruent triangles and the original triangle is isosceles.

Answer 3

Yes. The altitude is perpendicular to the base.
The median passes through the center of the base.
Since they are the same line, the line is the perpendicular bisector of the base. Thus it forms two right triangles with equal legs. Hence the triangles are congguent and the 3rd sides of the two right triangles are equal by CPCTC. Thus the triangle is isosceles.

Answer 4

Pretty straightfoward:
Let the triangle be ABC, B is the point at which the altitude is dropped to AC at point D.
angle ADB = angle CDB (right angles)
AD=DC (median)
BD=BD (idenity)
Triangle ABD = CBD (SAS=SAS)
AB=BC (corresponding props)
Triangles are isos (definition)