Question 16: To find the distance between the points \((-4, 0)\) and \((6, -8)\), we use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting in the values:
- \(x_1 = -4\)
- \(y_1 = 0\)
- \(x_2 = 6\)
- \(y_2 = -8\)
\[ d = \sqrt{(6 - (-4))^2 + (-8 - 0)^2} \]
This simplifies to:
\[ d = \sqrt{(6 + 4)^2 + (-8)^2} \] \[ d = \sqrt{10^2 + (-8)^2} \] \[ d = \sqrt{100 + 64} \] \[ d = \sqrt{164} \]
Calculating the square root:
\[ d \approx 12.81 \]
Rounding to the nearest tenth, the distance is 12.8.
Question 17: To find the midpoint of the line segment with endpoints \((0, 6)\) and \((-3, -3)\), we use the midpoint formula:
\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting in the values:
- \(x_1 = 0\)
- \(y_1 = 6\)
- \(x_2 = -3\)
- \(y_2 = -3\)
Calculating the coordinates of the midpoint:
\[ M = \left(\frac{0 + (-3)}{2}, \frac{6 + (-3)}{2}\right) \]
This simplifies to:
\[ M = \left(\frac{-3}{2}, \frac{3}{2}\right) = \left(-1.5, 1.5\right) \]
So the midpoint is (-1.5, 1.5).
Question 18: To find the other endpoint given one endpoint \((-4, 0)\) and the midpoint \((8, 3)\), we use the formula for the midpoint:
If \(M\) is the midpoint and \(A\) is one endpoint, then the other endpoint \(B\) can be found using:
\[ M = \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) \]
Solving for \(B\):
\[ M_x = \frac{x_A + x_B}{2} \quad \Rightarrow \quad 8 = \frac{-4 + x_B}{2} \quad \Rightarrow \quad 16 = -4 + x_B \quad \Rightarrow \quad x_B = 20 \]
\[ M_y = \frac{y_A + y_B}{2} \quad \Rightarrow \quad 3 = \frac{0 + y_B}{2} \quad \Rightarrow \quad 6 = y_B \]
Thus, the other endpoint is \((20, 6)\). Therefore, the student's answer is True, it is correct.
Final Answers:
- Question 16: 12.8
- Question 17: (-1.5, 1.5)
- Question 18: True
2 answers