Math Problem Statement

G is a circumcenter. Solve for x if BE = 9x - 10 and EC = 5x + 2.

Solution

To solve the problem, we need to determine the value of xx given the conditions:

  • BE=9x10BE = 9x - 10
  • EC=5x+2EC = 5x + 2

Key Information:

  1. G is the circumcenter:
    • In a triangle, the circumcenter is equidistant from all the vertices.
    • This implies that BE=ECBE = EC, as they are radii of the circumcircle.

Step-by-step Solution:

  1. Set the given expressions for BEBE and ECEC equal: 9x10=5x+29x - 10 = 5x + 2

  2. Simplify the equation: Subtract 5x5x from both sides: 4x10=24x - 10 = 2

    Add 1010 to both sides: 4x=124x = 12

  3. Solve for xx: Divide both sides by 4: x=3x = 3

Final Answer:

The value of xx is 3\boxed{3}.

Would you like me to verify the solution, or do you have additional questions?

Here are 5 related questions to expand your understanding:

  1. What is the significance of the circumcenter in a triangle?
  2. How do you find the circumcenter in different types of triangles?
  3. Can the circumcenter lie outside the triangle? If so, in what cases?
  4. What are some real-world applications of circumcenters in geometry?
  5. How would the problem change if the triangle were scalene or obtuse?

Tip: Always ensure your geometric conditions (like equidistant properties) are applied correctly when solving such problems!

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Math Problem Analysis

Mathematical Concepts

Geometry
Circumcenters
Linear Equations

Formulas

Equidistance property of a circumcenter in a triangle

Theorems

Circumcenter theorem

Suitable Grade Level

Grades 8-10