## Tuesday, July 5, 2011

### The triangle inequality, Heron's formula and the test for acute, right or obtuse using side lengths.

You probably already learned in high school that the interior angles of a triangle always add up to 180°.  Just as importantly, the Triangle Inequality tells us that any two side lengths of a triangle must add up to more than the third side length.  Another way to say this is that every side length must be less than half the total perimeter.

The idea of half the perimeter or semi perimeter shows up in Heron's Formula, a way to find the area of a triangle if you are given the three side lengths.

We also have a test using side lengths to see if a triangle is acute, right or obtuse.  If we have three side u, v and w and we declare that w is the long side, then we can use the sum of the squares of the short side to see how a triangle is classified.

u² + v² < w² : The triangle is obtuse

u² + v² = w² : The triangle is right

u² + v² > w² : The triangle is acute

Here are some practice problems for area and classification.

For each of these triples of numbers:
1) Determine if the triangle is equilateral, isosceles or scalene.
2) Determine if the triangle is acute, right or obtuse.
3) Find the area as a simplified square root
4) Find the area rounded to the nearest thousandth

a) 1, 1, 1
b) 1, 2, 2
c) 1, 3, 3
d) 1, 4, 4
e) 2, 2, 2
f) 2, 2, 3
g) 2, 3, 3
h) 2, 3, 4
i) 2, 4, 4
j) 3, 3, 3
k) 3, 3, 4
L) 3, 4, 4
m) 4, 4, 4

Prof. Hubbard said...

Step by step getting the answers for Heron's formula.

a) 1, 1, 1

perimeter = 3 semi-perimeter = 3/2
area = sqrt(3/2*(3/2-1)*(3/2-1)*(3/2-1))
= sqrt(3/2*1/2*1/2*1/2)
= sqrt(3/16)
= sqrt(3)/4

b) 1, 2, 2
perimeter = 5 semi-perimeter = 5/2
area = sqrt(5/2*(5/2 - 1)*(5/2 - 2)*(5/2 - 2))
= sqrt(5/2*3/2*1/2*1/2)
= sqrt(15/16)
= sqrt(15)/4

c) 1, 3, 3
perimeter = 7 semi-perimeter = 7/2
area = sqrt(7/2*(7/2 - 1)*(7/2 - 3)*(7/2 - 3))
= sqrt(7/2*5/2*1/2*1/2)
= sqrt(35/16)
= sqrt(35)/4

d) 1, 4, 4
perimeter = 9 semi-perimeter = 9/2
area = sqrt(9/2*(9/2 - 1)*(9/2 - 4)*(9/2 - 4))
= sqrt(9/2*7/2*1/2*1/2)
= sqrt(9*7/16)
= 3*sqrt(7)/4

e) 2, 2, 2
perimeter = 6 semi-perimeter = 3
area = sqrt(3*(3-2)*(3-2)*(3-2))
= sqrt(3*1*1*1)
= sqrt(3)

f) 2, 2, 3
perimeter = 7 semi-perimeter = 7/2
area = sqrt(7/2*(7/2 - 2)*(7/2 - 2)*(7/2 - 3))
= sqrt(7/2*3/2*3/2*1/2)
= sqrt(7*3*3/16)
= 3*sqrt(7)/4

g) 2, 3, 3
perimeter = 8 semi-perimeter = 4
area = sqrt(4*(4-2)*(4-3)*(4-3))
= sqrt(4*2*2*1)
= sqrt(8)
= 2*sqrt(2)

1) isosceles.
2) acute.
3) 2*sqrt(2)
4 2.828

h) 2, 3, 4
perimeter = 9 semi-perimeter = 9/2
area = sqrt(9/2*(9/2-2)*(9/2-3)*(9/2-4))
= sqrt(9/2*5/2*3/2*1/2)
= sqrt(9*15/16)
= 3*sqrt(15)/4

i) 2, 4, 4
perimeter = 10 semi-perimeter = 5
area = sqrt(5*(5-2)*(5-4)*(5-4))
= sqrt(5*3*1*1)
= sqrt(15)

j) 3, 3, 3
perimeter = 9 semi-perimeter = 9/2
area = sqrt(9/2*(9/2 - 3)*(9/2 - 3)*(9/2 - 3))
= sqrt(9/2*3/2*3/2*3/2)
= sqrt(9*9*3/16)
= 9*sqrt(3)/4

k) 3, 3, 4
perimeter = 10 semi-perimeter = 5
area = sqrt(5*(5-3)*(5-3)*(5-4))
= sqrt(5*2*2*1)
= 2*sqrt(5)

l) 3, 4, 4
perimeter = 11 semi-perimeter = 11/2
area = sqrt(11/2*(11/2 - 3)*(11/2 - 4)*(11/2 - 4))
= sqrt(11/2*5/2*3/2*3/2)
= sqrt(55*9/16)
= 3*sqrt(55)/4

m) 4, 4, 4
perimeter = 12 semi-perimeter = 6
area = sqrt(6*(6-4)*(6-4)*(6-4))
= sqrt(6*2*2*2)
= 4*sqrt(3)

Prof. Hubbard said...

a) 1, 1, 1
1) equilateral.
2) acute.
3) sqrt(3)/4
4) 0.433

b) 1, 2, 2
1) isosceles.
2) acute.
3) sqrt(15)/4
4) 0.968

c) 1, 3, 3
1) isosceles.
2) acute.
3) sqrt(35)/4
4) 1.479

d) 1, 4, 4
1) isosceles.
2) acute.
3) 3*sqrt(7)/4
4) 1.984

e) 2, 2, 2
1) equilateral.
2) acute.
3) sqrt(3)
4) 1.732

f) 2, 2, 3
1) isosceles.
2) obtuse.
3) 3*sqrt(7)/4
4) 1.984

g) 2, 3, 3
1) isosceles.
2) acute.
3) 2*sqrt(2)
4) 2.828

h) 2, 3, 4
1) scalene
2) obtuse
3) 3*sqrt(15)/4
4) 2.905

i) 2, 4, 4
1) isosceles.
2) acute.
3) sqrt(15)
4) 3.873

j) 3, 3, 3
1) equilateral.
2) acute.
3) 9*sqrt(3)/4
4) 3.897

k) 3, 3, 4
1) isosceles.
2) acute.
3) 2*sqrt(5)
4) 4.472

L) 3, 4, 4
1) isosceles.
2) acute.
3) 3*sqrt(55)/4
4) 5.562

m) 4, 4, 4
1) equilateral.
2) acute.
3) 4*sqrt(3)
4) 6.928