  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

 Nov 18, 2012 Heron’s Formula by: Staff AnswerPart I Plot each of the following points, and connect them to form the triangle ABC. : A(-2,5); B(1, 3); C(-1,0)  ----------------------------------------------------------------

 Nov 18, 2012 Heron’s Formula by: Staff ----------------------------------------------------------------Part IIFind 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 = 6.5Final Answer: Area = 6.5 ` ----------------------------------------------------------------

 Nov 18, 2012 Heron’s Formula by: Staff ----------------------------------------------------------------Part III ----------------------------------------------------------------

 Nov 18, 2012 Heron’s Formula by: Staff ----------------------------------------------------------------Part IVHeron's FormulaTo 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 ` ----------------------------------------------------------------

 Nov 18, 2012 Heron’s Formula by: Staff ----------------------------------------------------------------Part VLast, 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