D(-1,8), E(4,-2), F(-5,–3) are midpoints ofsides BC, CA and AB of triangle ABC . Find(i) equations of sides of triangle ABC(ii) co-ordinates of the circumcenter oftriangle ABC.
Answers
2x + y + 13 = 0 is Equation of AB & x - 9y + 73 = 0 is Equation of BC & 11x - 4y - 52 = 0 is Equation of AC
Step-by-step explanation:
Let say A (Ax , Ay) , B (Bx , By) & C (cx , Cy)
D(-1,8), E(4,-2), F(-5,–3) are midpoints of sides BC, CA and AB of triangle ABC
=> -1 = (Bx + cx)/2 & 8 = (By + Cy)/2
=> Bx + Cx = - 2 & By + Cy = 16
similarly
Ax + Cx = 8 & Ay + Cy = - 4
Ax + Bx = -10 & Ay + By = - 6
Adding all
=> 2(Ax + Bx + Cx) = -4 & 2(Ay + By + Cy) = 6
=> Ax + Bx + Cx = -2 & Ay + By + Cy = 3
Ax = 0 , Ay = - 13
Bx = -10 By = 7
Cx = 8 Cy = 9
A (0 , -13) , B (-10 , 7) , C (8 , 9)
AB slope = (7 - (-13)/(-10 - 0) = -2
y = -2x + c
-13 = 0 + c
=> y = -2x - 13 => 2x + y + 13 = 0 is Equation of AB
BC slope = 1/9
y = x/9 + c => 9 = 8/9 + c => c = 73/9
=> y = x/9 + 73/9 => 9y = x + 73 => x - 9y + 73 = 0 is Equation of BC
AC slope = 22/8 = 11/4
y = 11x/4 + c => -13 = 0 + c => c = -12
=> y = 11x/4 - 13 => 4y = 11x - 52 => 11x - 4y - 52 = 0 is Equation of AC
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Step-by-step explanation:
( ii) coordinates of the circumcenter of triangle ABC