What is the area of a triangle given (2, 3), (4, 5), and (6, 7)?

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SUMMARY

The area of a triangle formed by the points (2, 3), (4, 5), and (6, 7) is zero, as these points are collinear. The area can be calculated using the formula A = (1/2)[(ad - cb) + (cf - ed) + (eb - af)], but in this case, the result confirms the collinearity. For those unfamiliar with direct area calculation, applying the distance formula three times and utilizing Heron's Formula are recommended methods to explore. A tutorial on deriving the area formula for three points in a plane is available for further reference.

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  • Understanding of coordinate geometry
  • Familiarity with the distance formula
  • Knowledge of Heron's Formula
  • Basic algebraic manipulation skills
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  • Study the derivation of the area formula for a triangle given three points in a plane
  • Learn how to apply the distance formula in coordinate geometry
  • Explore Heron's Formula for calculating the area of triangles
  • Investigate the concept of collinearity in geometry
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Given (2, 3), (4, 5), (6, 7), find the area using the formula below.

Note: (a, b) (c, d) (e, f)

a = 2
b = 3

c = 4
d = 5

e = 6
f = 7

A = (1/2)[(ad - cb) + (cf - ed) + (eb - af)]

Just plug and chug, right?
 
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Suppose you weren't given a formula into which you may "plug and chug"...can you outline a method you could use to find the area?
 
1. Apply distance formula 3 times.

2. Heron's Formula
 
Given (2, 3), (4, 5), (6, 7), find the area using the formula below.

Note: (a, b) (c, d) (e, f)

a = 2
b = 3

c = 4
d = 5

e = 6
f = 7

A = (1/2)[(ad - cb) + (cf - ed) + (eb - af)]

the area will be zero ... those three points are collinear
 
MarkFL said:
Suppose you weren't given a formula into which you may "plug and chug"...can you outline a method you could use to find the area?

1. Apply distance formula 3 times.

2. Heron's Formula.
 
RTCNTC said:
1. Apply distance formula 3 times.

2. Heron's Formula.

Yes, that's likely the quickest way to get the area without an explicit formula for the area of a triangle formed by 3 points in the plane. (Yes) I once posted a tutorial here on deriving a general formula:

http://mathhelpboards.com/math-notes-49/finding-area-triangle-formed-3-points-plane-2954.html

And it relies on the following derivation:

http://mathhelpboards.com/math-notes-49/finding-distance-between-point-line-2952.html
 
Great information.
 

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