Question

In a triangle abc if ad=3 cm; ab=12 cm, ae= 4 cm ed=16 cm. check that de parallel to ab

Answer 1

A perpendicular bisector is a line which cuts a line segment into two equal parts at 90°. So AC to be a perpendicular bisector of BD it must not only cut it at 90° (which it does) but also cut it into two equal parts. Now, in order AC to cut BD into two equal parts right triangle ABD must be isosceles, which, as it turns out after some math, it is not.

Complete solution:
In triangle ABC, if BC = 3 and AC = 4, then what is the length of segment CD?
A. 3 
B. 15/4
C. 5 
D. 16/3 
E. 20/3



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Important property: perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.

Thus, the perpendicular AC divides right triangle ABD into two similar triangles ACB and DCA (which are also similar to big triangle ABD). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles marked with red and blue on the diagram). 

So, CDAC=ACBCCDAC=ACBC --> CD4=43CD4=43 --> CD=163CD=163.

Answer: D.