Math Problem Statement

Find the missing endpoint of a segment with one endpoint at (-8, -5) and the midpoint at (2, 3).

Solution

To find the missing endpoint of a segment when one endpoint and the midpoint are given, we can use the midpoint formula:

M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Given:

  • One endpoint A(8,5)A(-8, -5)
  • Midpoint M(2,3)M(2, 3)
  • Missing endpoint B(x2,y2)B(x_2, y_2)

We can set up the following system of equations based on the midpoint formula:

8+x22=2(for the x-coordinate)\frac{-8 + x_2}{2} = 2 \quad \text{(for the x-coordinate)} 5+y22=3(for the y-coordinate)\frac{-5 + y_2}{2} = 3 \quad \text{(for the y-coordinate)}

Solving for x2x_2:

8+x22=2\frac{-8 + x_2}{2} = 2 Multiply both sides by 2: 8+x2=4-8 + x_2 = 4 Solve for x2x_2: x2=4+8=12x_2 = 4 + 8 = 12

Solving for y2y_2:

5+y22=3\frac{-5 + y_2}{2} = 3 Multiply both sides by 2: 5+y2=6-5 + y_2 = 6 Solve for y2y_2: y2=6+5=11y_2 = 6 + 5 = 11

Thus, the missing endpoint is B(12,11)B(12, 11).

The correct answer is (12,11)\boxed{(12, 11)}.

Would you like any further clarification or details on how this was calculated?

Here are 5 related questions:

  1. How do you calculate the midpoint if both endpoints are known?
  2. How does the midpoint formula apply in 3D geometry?
  3. What is the significance of a midpoint in dividing a line segment?
  4. Can the midpoint formula be used for weighted averages in statistics?
  5. How would you find the midpoint if one endpoint and the midpoint are fractions?

Tip: Always double-check your calculations, especially when solving for unknowns, to avoid simple arithmetic mistakes!

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Midpoint Formula
Linear Equations

Formulas

Midpoint Formula: M = ((x1 + x2) / 2, (y1 + y2) / 2)

Theorems

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Suitable Grade Level

Grades 8-10