An equilateral triangle has two vertices at the points (3, 4) and (-2, 3), find the coordinates of the third vertex.
Sol:
Two vertices of an equilateral triangle are (3, 4) and (-2, 3)
Let the third vertex of the triangle be (x, y)
Distance between (3, 4) and (-2, 3)
=√(−2−3)2+(3−4)2
=√(−5)2+(−1)2
⇒ √26
Distance between (3, 4) and (x, y)
= √[(x−3)2+(y−4)2]=√x2−6x+9+y2−8y+16=√x2−6x+y2−8y+25 ------------------------ (1)
Distance between (-2, 3) and (x, y)
= √(x+2)2+(y−3)2=√(x+2)2+(y−3)2=√x2+4x+4+y2−6y+9=√x2+4x+y2−6y+13-------------------- (2)
Equating the distances we get,
x2−6x+y2−8y+25=x2+4x+y2−6y+13
10x + 2y - 12 = 0
5x + y - 6 = 0
y = (6 - 5x)
Substituting the value of y in equation (1) and equating it to √26 snd squaring on both sides we get
x2−6x+y2−8y+25=26⇒x2−6x+(6−5x)2−8(6−5x)+25=26⇒x2−6x+36+25x2−60x−48+40x+25=26⇒26x2−26x−13=0⇒2x2−2x−1=0
Solving the quadratic equation using the quadratic formula, −b±√b2−4ac2a
x = 2±√4+84
x = 2±√124
x = 2±2√34
x = 1±√32
y = (6−5x)=6−5(1±√32)=12−5±5√32=7±5√32
Hence, the coordinates of the third vertex of the equilateral triangle are [1±√3]2,7±5√32.