During a math exam, one of the students erroneously copied a trig problem, the original being:

In a triangle ABC, AB=9 cm, AC="...", and the angle BAC=60 deg.

Find the length of side BC.
"..." provided a number which the student copied, increasing it by 1 cm.

Still he got the correct answer.

What was it?

I get BC~7.81 cm.

Method: I made the base AC and imagined angle ACB to be slightly great than 90 deg. I then extended the base to erroneous point C', 1 cm further out. Since BC=BC', triangle BCC' is isosceles. I dropped a perpendicular from B to the base, which I called D which is halfway between C and C'. CD = C'D = 0.5 cm.

Now, BD = 9 sin 60 = 9 sqrt(3)/2. The similar right triangles BDC and BDC' have sides 0.5 and 9 sqrt(3)/2, and therefore a hypotenuse:

BC = BC' = sqrt { (1/2^2 + [9 sqrt(3)/2 ]^2 } = 7.81 cm

The number, miscopied was AC = AD -0.5 = AB cos 60 -.05

= 9 x 1/2 - 0.5 = 4 cm. He wrote 5 cm.

*Edited on ***August 30, 2018, 1:22 pm**