Prove the converse of Theorem 9-26: In a triangle, if a median is half as long as the side which it bisects,?

Question

then the triangle is a right triangle and the side is its hypotenuse. Given: triangle ABC, median segment AD, AD=1/2 BC. Prove Triangle ABC is a right triangle, and segment BC is its hypotenuse. Hint: Prove x+y=90. thank you...

Answer 1

There are two isosceles triangles.
Therefore Angle A = x, Angle B = y
x + y = C = A + B
A + B + C = 180
=> 2 *C = 180.

Answer 2

i'm going to anticipate devoid of loss of generality that AB >= CD and that i hire the notation := to define a variable. here is yet another technique to coach the median is parallel to the different 2 bases making use of comparable triangles. First enhance CD and BC to fulfill at O. notice that ODC ~ OAB by AA similarity. permit OD = a, OC = b, DE = EA = x, and CF = facebook = y. From an analogous triangles we've here relationship (2x + a) / a = (2y + b) / b.....(a million) If (a million) implies (a + x) / a = (b + y) / b, then OEF ~ ODC and the effect follows at present. permit's control (a million): (2x + a) / a = (2y + b) / b (2x/a) + a million = (2y/b) + a million 2x/a = 2y/b x/a = y/b (x/a) + a million = (y/b) + a million (x + a) / a = (y + b) / b and we are carried out. to coach EF = (AB + CD)/2 i might propose a fashion equivalent to Holdm's. Drop perpendiculars from D to EF intersecting at G and from E to AB intersecting at H. We notice that DEG and EAH are congruent by AAS congruency. as a effect AH = EG :=m. Analogously appearing an analogous operation alongside CB with perpendiculars CI and FJ we've FI = BJ :=n. Then DC = EF - (m + n) and AB = EF + (m + n). for sure EF is the conventional of AB and CD. the attitude above would be prolonged for the area data. If we rotate triangle EAH a hundred and eighty stages clockwise approximately E and rotate triangle FBJ a hundred and eighty stages counterclockwise approximately F we style a rectangle with one length equivalent to the peak and one length equivalent to the median. this might lend right to a data devoid of words style diagram. Addition: Denis, i think of of an stylish answer as one that is easy and insightful.

Answer 3

............A ^
............./x \y\
............/.....\...\
.........../........\...\
........../..........\.....\
........./_______\__\
.......B............D.......C
I drew the figure. Rest you find out. You must do math yourself

Answer 4

no.