If angle BAD is a right angle, what is the length of side

Isn't it that in an right angle triangle a perpenidcular line to hupothenuses (BC) halves the 90 degree triangle into 45° and 45° degrees?

Hi Bunuel,

Small question here

Since the AC divides the triangle into three similar triangles. and we have side AD as 5. Cant we calculate ACD as a pyth triplet, hence 3-4-5 and then
AC/BC=CD/AC. This would give us BC??

For such kind of graphic questions you MUST post the image. Next, please also do check the OA's when posting a question, OA for this one is C, not E.

Original question is below:

If angle BAD is a right angle, what is the length of side BD?


(1) AC is perpendicular to BD
(2) BC = CD

Now, obviously each statement alone is not sufficient.

When taken together we have that AC is a perpendicular bisector. Now, if a line from a vertex to the opposite side is both perpendicular to it and bisects it then this side is a base of an isosceles triangle (or in other words if a bisector and perpendicular coincide then we have an isosceles triangle). You can check this yourself: in triangles ACD and ACB two sides are equal (AC=AC and BC=CD) and included angle between these sides are also equal (<ACD=<ACB=90) so we have Side-Angle-Side case (SAS), which means that ACD and ACB are congruent triangles, so AB=AD --> ABD is an isosceles triangle.

Next, as ABD is an isosceles triangle then AB=AD=5 and hypotenuse \(BD=5\sqrt{2}\).

Answer: C.

For more on this Triangles chapter of Math Book: https://gmatclub.com/forum/math-triangles-87197.html

Hope it helps.

Yes, the perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle. But the corresponding (equal) angles would be BAC and ADC (also DAC and ABC) not CDA and ABC.

Hope it's clear.