Cartesian Plane, SMB-020
Prove the points `(1,-1), (-1,1)` and `(-sqrt3,-sqrt3)` are the vertices of a equilateral triangle. (4 marks)
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An equilateral triangle has vertices `O(0,0)` and `A(8,0)` as shown in the diagram below.
Find `k` if the coordinates of the third vertex are `B(4,k)`. (4 marks)
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`text{Proof (See worked solutions)}`
`ΔOAB\ text{is equilateral}\ \ =>\ \ OA=AB=OB=8`
`text{Let}\ C=(0,4)`
`text{Consider}\ ΔOCB:`
`OB^2` | `=OC^2+CB^2` | |
`64` | `=16+CB^2` | |
`CB^2` | `=48` | |
`CB` | `=sqrt(48)` | |
`=4sqrt(2)` |
`B=(4,4sqrt(2))`
`:.k=4sqrt(2)`
Prove the points `(1,-1), (-1,1)` and `(-sqrt3,-sqrt3)` are the vertices of a equilateral triangle. (4 marks)
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`text{Proof (See worked solutions)}`
`text{Let points be:}\ A(1,-1), B(-1,1) and C(-sqrt3,-sqrt3)`
`text(Using the distance formula):`
`d_(AB)` | `=sqrt{(x_2-x_1)^2+(y_2-y_1)^2}` | |
`=sqrt{(1-(-1))^2+(-1-1)^2}` | ||
`=sqrt8` |
`d_(BC)` | `=sqrt{(-1-(-sqrt3))^2+(1-(-sqrt3))^2}` | |
`=sqrt{(-1+sqrt3)^2+(1+sqrt3)^2}` | ||
`=sqrt(1-2sqrt3+3 +1+2sqrt3+3)` | ||
`=sqrt8` |
`d_(AC)` | `=sqrt{(1-(-sqrt3))^2+(-1-(-sqrt3))^2}` | |
`=sqrt{(1+sqrt3)^2+(-1+sqrt3)^2}` | ||
`=sqrt(1+2sqrt3+3 +1-2sqrt3+3)` | ||
`=sqrt8` |
`text{Since}\ AB=BC=AC`
`:. ΔABC\ text{is equilateral.}`