Determine third point of triangle when two points and all sides are known?

Question

Determine third point of triangle (on a 2D plane) when two points and all sides are known? A = (0,0) B = (5,0) C = (?, ?) AB = 5 BC = 4 AC = 3 Can someone please explain how to go about this? I understand there will be two possible points and would like to arrive at both. This is what I've worked out but I'm uncertain at this point how correct it is. C.x = (AB² - BC² + AC²) / (2 * AB) C.y = sqrt(BC² - (B.x - C.x)²) - B.y Thanks! **Update - Need to turn the answer into a reusable formula, solving for C.x and C.y** known sides AB, BC, AC known points A(x, y), B(x, y) unknown points C(x, y) AC² - BC² = ((Ax - Cx)² + (Ay - Cy)²) - ((Bx - Cx)² + (By - Cy)²) Goal: C.x = ? C.y = ?

Answer 1

I designated C (x,y)

The distance of AC using distance formula is

3= sqrt (x^2 + y^2 )

squaring both sides 9 = x^2 + y^2

y^2=9-x^2

The distance of CB

4=sqrt[(x-5)^2 + y^2]

square both sides

16=(x-5)^2+y^2

substitute y^2=9-x^2

16=(x-5)^2 + 9 - x^2

Foil

16= x^2 -10x +25 + 9 - x^2

Combine like terms

16=-10x +34

Subtract 34 from both sides

-18=-10x

Divide Both sides by -10

x=1.8

y=sqrt (9-(1.8)^2)=2.4

You can check this. The angle at C is 90 so the slopes of

AC and BC must be negative reciprocals.

Slope AC (2.4-0)/(1.8-0)=4/3

Slope BC (2.4-0)/(1.8-5)= -3/4

Point C has the coordinates (1.8, 2.4)

Answer 2

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