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.

*u*² +

*v*² <

*w*² : The triangle is obtuse

*u*² +

*v*² =

*w*² : The triangle is right

*u*² +

*v*² >

*w*² : 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

Answers in the comments.

## 2 comments:

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)

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

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