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Question #5676
What is the measurement of AB? abc is an isosceles triangle with ab = ac and a median (bd) the perimter of abc is 60 units . what is the measurment of segment ab
We suppose that AB=AC=BD(median), because if not then there is no use for BD, and the information is not sufficient
1) AB=AC=BD=x
2) BD is a median ⟹ according to the definition of the median, AD=DC=x/2 (Look at the image1)
3) Let’s construct a BH –an altitude in the triangle ABD. According to the definition of the altitude BH is perpendicular to the vertex AD n to the vertex AC, because AC includes AD. (Look at the image1). So angle ∠BHA=∠BHD=90˚ and we get two right triangles – BHA and BHC.
4)ABD is an isosceles triangle, because AB=AD=x. In the isosceles triangle altitude BH is also a median of the triangle ABD ⟹ AH =HD =(x/2)/2=x/4.
5) Theorem by Pythagoras defines the relationship between the three sides of a right triangle: , where& is the length of the hypotenuse and , are the lengths of the other two sides.
6) According to this theorem, use it to the triangle AHB(where AB=x and AH=x/4) then we can find
7) According to this theorem, use it to the triangle BHC(where HC=HD +DC = x/4 + x/2 =3/4x and
& then we can find
<img style="width: 258px; height: 30px;" src="data:image/png;base64,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" alt="">
8) So
9) The& perimeter of ABC is 60 units, so
10) From that we can find
<img style="width: 115px; height: 51px;" src="data:image/png;base64,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" alt="">

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