How long is it's longest side?

Question

A triangle has area 72cm^2, and two of its angels measure 44degrees and 62 degrees. How long is it's longest side?

Answer 1

Use the law of sines:

http://www.mathopenref.com/lawofsines.html

 

Note that the 3rd angle is 180 - (44 + 62) = 54 degrees

 

If the sides are a, b, and c, we can now write:

a/sin 44 = b/sin 62 = c/sin 54

 

Note that b will be the longest side.  To develop the equation for the area, let b be the base.  To relate to the area, we now need the height.

 

Draw the height as a line perpendicular to b--call the 2 segments b1 (nearest the 44 deg angle) and b2 (nearest the 54)

 

Now write two equations relating the height to the 2 segments:

h/b1 = tan 44

h/b2 = tan 54

 

Rearrange to get h as a function of b:

b1 = h/tan44

b2 = h/tan 54

therefore:  b = h/tan44 + h/tan54

 

Using the area formula, write this:

 

(b * h)/2 = A = 72cm^2

rearrange to be a function of h:

b * h = 144

 

h =144/b

 

Finally, substitute into the formula for b:

 

b = (144/b)/tan44 + (144/b)/tan54

multiply both sides by b

b^2 = 144 ( 1/tan44 + 1/tan54)

 

evaluate:

b^2 = 144 ( 1.036 + 0.7265 ) = 253.8

 

b = 15.93 cm

 

CHECK:

h = 144 / b = 144/15.93 = 9.039

 

area:  A = b * h /2 = 15.93 * 9.039 / 2 = 72