Question

for questions 1 to 3, choose the correct answer. 1. the mid - point of the line segment with endpoints a(-3, -3) and b(1, 5) is at a (-2, 2) b (-4, -8) c (-1, 1) d (1, -1) 2. the length of the line segment with endpoints c(-5, 2) and d(1, -4) is a √20 b √24 c √72 d √80 3. an equation for the circle with centre (0, 0) and radius 4 is a x² + y² = 2 b x² + y² = 4 c x² + y² = 8 d x² + y² = 16

Explanation:

1. Mid - point formula:

• The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
• Given \(A(-3,-3)\) and \(B(1,5)\), then \(x_1=-3,y_1 = - 3,x_2 = 1,y_2 = 5\).
• Calculate the \(x\) - coordinate of the mid - point: \(\frac{-3 + 1}{2}=\frac{-2}{2}=-1\).
• Calculate the \(y\) - coordinate of the mid - point: \(\frac{-3 + 5}{2}=\frac{2}{2}=1\).
• The mid - point is \((-1,1)\).

1. Distance formula:

• The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• Given \(C(-5,2)\) and \(D(1,-4)\), then \(x_1=-5,y_1 = 2,x_2 = 1,y_2=-4\).
• First, calculate \((x_2 - x_1)\) and \((y_2 - y_1)\): \(x_2 - x_1=1-(-5)=6\) and \(y_2 - y_1=-4 - 2=-6\).
• Then, \(d=\sqrt{(6)^2+(-6)^2}=\sqrt{36 + 36}=\sqrt{72}\).

1. Equation of a circle:

• The standard form of the equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\).
• Given the center \((0,0)\) and \(r = 4\), substituting \(h = 0,k = 0,r = 4\) into the formula, we get \(x^{2}+y^{2}=16\).

Answer:

1. C. \((-1,1)\)
2. C. \(\sqrt{72}\)
3. D. \(x^{2}+y^{2}=16\)