Question

Consider the points A(0,8),B(−6,0) and C(6,0). Let L₁ and L₂ be the perpendicular bisectors of the segments AB and AC, respectively, and let T be the intersection point of the lines L₁ and L₂. Let C denote the circle which is centred at T, and passes through B.

(a) Find the coordinates of the point T.

Answer 1

The coordinates of point T, we need to determine the equations of perpendicular bisectors L₁ and L₂ of segments AB and AC, then solve for their intersection. The coordinates of T, where these bisectors intersect, are (0, 4).

Step-by-step explanation:

To find the coordinates of the point T, which is the intersection of the perpendicular bisectors L₁ and L₂ of the segments AB and AC, we first need to determine the equations of these lines. A perpendicular bisector of a segment will have a slope that is the negative reciprocal of the segment's slope and passes through the segment's midpoint.

The midpoint of AB (0,8) and B(-6,0) can be found by averaging their coordinates:

M1 = ((0-6)/2, (8+0)/2) = (-3, 4)

The slope of AB is (0-8)/(-6-0) = 4/3, so the slope of L₁ is the negative reciprocal, -3/4. Now, using the point-slope form, the equation of L₁ is

y - 4 = -3/4(x + 3).

Similarly, the midpoint of AC (0,8) and C(6,0) is

M2 = ((0+6)/2, (8+0)/2) = (3, 4)

The slope of AC is (0-8)/(6-0) = -4/3, making the slope of L₂ the negative reciprocal, 3/4. The equation of L₂ is

y - 4 = 3/4(x - 3).

Solving for the intersection of L₁ and L₂, we set the right sides of both equations equal to each other and solve for x:

-3/4(x + 3) = 3/4(x - 3)

Solving this gives x = 0. Substituting x into one of the equations gives y = 4. Thus, the coordinates of T are (0, 4).