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Type: Open-Ended
Category: Triangles
Level: Grade 11
Author: nsharp1
Created: 4 years ago

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Triangles Question

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For [math]Delta ABC[/math], as seen below, let the intersection of the altitude from vertex [math]B[/math] and side [math]bar{AC}[/math] be point [math]D[/math] (not labeled). Also, let [math]b = AC, \ c=AB, \ a = BC, \ x = AD, \ y=CD[/math], and [math]h = BD[/math]. Finally, let [math]P = a+b+c[/math] and [math]s=P/2[/math].


The area of this triangle can be expressed using Heron's formula, [math]A_{Delta ABC} = sqrt(s(s-a)(s-b)(s-c))[/math], where [math]s = (a+b+c)/2[/math]. The following questions will derive this formula.

Acute Triangle Height v2

Grade 11 Triangles

Using the final step from the previous question, and then taking the square root of both sides of the equation yields: [math]h = 1/(2b)sqrt(P(P - 2a)(P - 2b)(P - 2c))[/math]. This can be substituted into the equation for the area of [math]Delta ABC[/math] found in the first question. Do this, and incorporate the constant factor into the square root. This should enable one to substitute in the value [math]s[/math] and result in Heron's formula.