## Monday, May 13, 2019

### Sides of an Isosceles Triangle

This geometry problem asks you to find the length of each side of an Isosceles triangle ABC where AB=BC given the perimeter of the triangle and the angle ABC.  I ask you to set up the solution in an Excel spreadsheet where triangle perimeter and length of angle ABC are inputs, which can be modified by user of spreadsheet.

Question One:   You form an Isosceles triangle ABC with 12 inches of string.   You know AB=BC and you denote the length of both sides X.  Set up a spreadsheet where you can find the length of all sides of the triangle for different values of angle ABC.

Answer:  We know that AB+BC+AC =12.  Denote AB and BC as X.  Denote the angle ABC as a.   We know that half of BC is X*sin(a/2) so BC is 2*X*sin(a/2).

this means 2X  +2sin(a/2)X =12

rearrange to get x

X  = 12/(2*(1+sin(a/2))

The table below is from an Excel spreadsheet calculating triangle side length for several different values of angle ABC.

 Calculating Isosceles Triangle Sides String Length 12 12 12 12 12 12 Angle between equal sides a 30 45 60 90 120 150 sine a/2 0.26 0.38 0.5 0.71 0.87 0.97 x (AB or BC) 4.77 4.34 4 3.51 3.22 3.05 Hypotenuse BC 12-2x 2.47 3.32 4 4.97 5.57 5.90

Technical Note:  In Excel get sine from sin function and use radian function to translate degrees to radian.  sin(radians(30/2)) is equal to 0.26.

Intuitive Note:  We know that an Isosceles triangle with top angle of 60 degrees is an equilateral triangle.   Our calculations reveal that this is indeed the case.

Follow-up problem:  Find the area of all triangles listed in the chart above.

The height of the triangle (the line from point B to the midpoint of  AC) is H=X*Cos(a/2).

The area is 0.5*H*B where B is the base or in this case the hypotenuse.

Calculations are presented below.

 Calculating Isosceles Triangle Sides String Length 12 12 12 12 12 12 Angle between equal sides a 30 45 60 90 120 150 sine a/2 0.26 0.38 0.5 0.71 0.87 0.97 x (AB or BC) 4.77 4.34 4 3.51 3.22 3.05 Hypotenuse BC 2.47 3.32 4 4.97 5.57 5.90 Triangle Height 4.60396 4.0091 3.464 2.485 1.6077 0.7899 Area Height * Hypotenuse * 0.5 5.68 6.66 6.93 6.18 4.48 2.33

Intuitive Note:  Note that the equilateral triangle has the largest area.   This makes sense because the areal of an equilateral triangle is half the area of square.