Math Problem Statement

Find the midpoint and distance between (7,-3) and (5,2)

Solution

To find the midpoint and distance between the points (7,3)(7, -3) and (5,2)(5, 2), we use the following formulas:

Midpoint Formula:

The midpoint MM between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by: M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Substitute the coordinates (x1,y1)=(7,3)(x_1, y_1) = (7, -3) and (x2,y2)=(5,2)(x_2, y_2) = (5, 2): M=(7+52,3+22)=(122,12)=(6,0.5)M = \left( \frac{7 + 5}{2}, \frac{-3 + 2}{2} \right) = \left( \frac{12}{2}, \frac{-1}{2} \right) = (6, -0.5)

Distance Formula:

The distance dd between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by: d=(x2x1)2+(y2y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Substitute the coordinates (x1,y1)=(7,3)(x_1, y_1) = (7, -3) and (x2,y2)=(5,2)(x_2, y_2) = (5, 2): d=(57)2+(2+3)2=(2)2+(5)2=4+25=29d = \sqrt{(5 - 7)^2 + (2 + 3)^2} = \sqrt{(-2)^2 + (5)^2} = \sqrt{4 + 25} = \sqrt{29}

Thus, the distance is 29\sqrt{29} or approximately 5.395.39.

Final Answer:

  • Midpoint: (6,0.5)(6, -0.5)
  • Distance: 295.39\sqrt{29} \approx 5.39

Would you like more details or have any questions?


Related Questions:

  1. What is the formula for finding the slope between two points?
  2. Can you find the equation of the line passing through the points (7,3)(7, -3) and (5,2)(5, 2)?
  3. How do you find the distance between a point and a line?
  4. What is the general equation for the circle passing through two points?
  5. How can you find the centroid of a triangle with given vertices?

Tip: The distance formula is derived from the Pythagorean theorem, which connects the horizontal and vertical distances between two points.

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Math Problem Analysis

Mathematical Concepts

Midpoint
Distance between two points
Coordinate Geometry

Formulas

Midpoint Formula: M = ((x1 + x2)/2, (y1 + y2)/2)
Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 7-9