The figure shows a triangular wooden frame ABC. The side AD of the frame has rotted and needs to be replaced.

Question

anonymous · Answer

What is the length of the wood that is needed to replace AD?

8.1 inches

6.2 inches

5.9 inches

7.4 inches

anonymous · Answer

5.992

anonymous · Answer

how?

anonymous · Answer

tan30=DB/BC and tan45=AB/BC.
 tan45=1 and tan30=1/sqrt(3).
 tan45-tan30=1-1/sqrt(3)=0.423. 
Now, Substituting tan45 as AB/BC and tan30 as DB/BC.
Therefore, (AB-DB)/BC = 0.423
But, AB-DB =AD and by multiplying BC on both sides,
we get, AD = 0.423*BC , i.e. (0.423*14)inches = 5.922.
Hope thats not confusing :)

anonymous · Answer

no not at all, can you help me with another question.

kamille · Answer

You know, that AB=BC (because there is angle ACB=45degree, BAC=45degree). That's mean that the triangle is isosceles). So, AB=BC=14in.
Also, let's say that BD is x, than DC=2x (because the hypotenuse is twice the length of the shorter side)
Finally, acorrding to the Pythagorean theorem:
$$x ^{2}+14^{2}=(2x)^{2}$$
$$x ^{2}+196=4x ^{2}$$
$$3x ^{2}=196|:3$$
$$x \approx8.0.8$$
AD=AB-BC=14-8.08=5.91

anonymous · Answer

thanks so much :)

anonymous · Answer

@Kamille  Thats much simpler and better.