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Area of Triangle – Heron’s Formula

by Michelle
(Fayetteville, NC, USA)











































Heron’s Formula, Three-Point Coordinate Formula

   • Plot each of the following points, and connect them to form the triangle ABC.

           A(-2,5); B(1, 3); C(-1,0)

   • Find the area of the triangle using two techniques:

           Heron's Formula

           The three-point coordinate formula (see Math Open Reference, 2009).

Show your work in detail. Prove that both methods yield the same results.

I plotted the points on the x,y axis but I am lost when it comes to the Heron formula or the three point coordinate formula. For side a, I got √13 for side b, I got √26 and for side c is got 3. This is getting me nowhere. I really need help

Comments for Area of Triangle – Heron’s Formula

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Nov 18, 2012
Heron’s Formula
by: Staff


Answer

Part I


Plot each of the following points, and connect them to form the triangle ABC. : A(-2,5); B(1, 3); C(-1,0)




Triangle with vertices: A (-2,5);  B (1, 3);  C (-1,0)





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Nov 18, 2012
Heron’s Formula
by: Staff


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Part II


Find the area of the triangle A-B-C

Three-Point Coordinate Formula


       │ Ax(By - Cy) + Bx(Cy - Ay) + Cx(Ay - By) │
Area = │───────────────────────────────────────│
│ 2 │

Ax= the x coordinate of point A
Ay= the y coordinate of point A

Bx= the x coordinate of point B
By= the y coordinate of point B

Cx= the x coordinate of point C
Cy= the y coordinate of point C

A(-2,5); B(1, 3); C(-1,0)


│ (-2)(3 - 0) + 1 (0 - 5) + (-1)(5 - 3) │
Area = │───────────────────────────────────────│
│ 2 │

│ (-2)(3) + 1 (- 5) + (-1)(2) │
Area = │─────────────────────────────│
│ 2 │

│ (-6) + (- 5) + (-2) │
Area = │─────────────────────│
│ 2 │

│ -13 │
Area = │─────│
│ 2 │

Area = │- 6.5│

Area = 6.5

Final Answer:

Area = 6.5





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Nov 18, 2012
Heron’s Formula
by: Staff


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Part III




Triangle with sides: a, b, and c





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Nov 18, 2012
Heron’s Formula
by: Staff


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Part IV


Heron's Formula

To use Heron’s Formula, you must first calculate the length of sides a, b, and c.

   a = √[(Bx - Cx)² + (By - Cy)²]   

b = √[(Cx - Ax)² + (Cy - Ay)²]

c = √[(Bx - Ax)² + (By - Ay)²]


A(-2,5); B(1, 3); C(-1,0)



a = √[(1 – (-1)² + (3 - 0)²]

a = √[(1 + 1)² + (3)²]

a = √[(2)² + (3)²]

a = √(4 + 9)

a = √(13)

a ≈ 3.61



b = √[(-1 - (-2))² + (0 - 5)²]

b = √[(-1 + 2)² + (-5)²]

b = √[(1)² + (-5)²]

b = √(1 + 25)

b = √(26)

b ≈ 5.10



c = √[(1 – (-2))² + (3 - 5)²]

c = √[(1 + 2)² + (-2)²]

c = √[(3)² + (-2)²]

c = √(9 + 4)

c = √(13)

c ≈ 3.61


Second, calculate “S” (½ the length of the perimeter)


        a + b + c 
S = ───────────
2

a ≈ 3.61; b ≈ 5.10; c ≈ 3.61


3.61 + 5.10 + 3.61
S ≈ ────────────────────
2

12.32
S ≈ ─────
2

S ≈ 6.16





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Nov 18, 2012
Heron’s Formula
by: Staff

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Part V

Last, calculate the area of the triangle using Heron’s Formula

             ______________________ 
A = √S(S – a)(S – b)(S – c)

S ≈ 6.16; a ≈ 3.61; b ≈ 5.10; c ≈ 3.61


___________________________________________
A ≈ √6.16(6.16 – 3.61)(6.16 – 5.10)(6.16 – 3.61)
_______________________
A ≈ √6.16(2.55)( 1.06)( 2.55)
_______
A ≈ √42.4587

A ≈ 6.52

Final Answer:

Area = 6.5





Thanks for writing.

Staff
www.solving-math-problems.com



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