Point \(P(-4,2)\) lies on the line segment joining \(A(-4,6)\) and \(B(-4,-6)\).
True.
Step 1: If a point lies on the line segment joining two points, then the three points are collinear. In that case, the area of the triangle formed by them is zero.
Step 2: Use the area of triangle formula for points \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(P(x_3,y_3)\):
\(\text{Area} = \dfrac{1}{2}\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right|\).
Step 3: Substitute \(A(-4,6)\), \(B(-4,-6)\), \(P(-4,2)\).
Step 4: Compute inside the absolute value:
\(x_1(y_2-y_3) = (-4)\big((-6)-2\big) = (-4)(-8) = 32\)
\(x_2(y_3-y_1) = (-4)\big(2-6\big) = (-4)(-4) = 16\)
\(x_3(y_1-y_2) = (-4)\big(6-(-6)\big) = (-4)(12) = -48\)
Step 5: Add them:
\(32 + 16 - 48 = 0\)
Step 6: Area:
\(\text{Area} = \dfrac{1}{2}|0| = 0\)
Final Step: Since the area of \(\triangle ABP\) is zero, points A, B, and P are collinear. Therefore, \(P(-4,2)\) lies on the line segment joining A and B (its y-value 2 is between -6 and 6).