Math Problem Statement

If B is the midpoint of AC, and AC = 8x - 20, find BC. AB = 3x - 1.

Solution

In this problem, we are given the following information:

  • BB is the midpoint of ACAC.
  • AC=8x20AC = 8x - 20.
  • AB=3x1AB = 3x - 1.
  • We are asked to find BCBC.

Step-by-Step Solution

Since BB is the midpoint of ACAC, it divides the line segment into two equal parts. This means: AB=BCAB = BC

Also, since AC=AB+BCAC = AB + BC, we can express BCBC as BC=ABBC = AB. So: AC=2×ABAC = 2 \times AB

Substituting the values we know: 8x20=2×(3x1)8x - 20 = 2 \times (3x - 1)

Solve for xx:

8x20=6x28x - 20 = 6x - 2

Now, simplify: 8x6x=2+208x - 6x = -2 + 20 2x=182x = 18 x=9x = 9

Find BCBC:

Since AB=BCAB = BC and AB=3x1AB = 3x - 1, substitute x=9x = 9 into the expression for ABAB: AB=3(9)1=271=26AB = 3(9) - 1 = 27 - 1 = 26 So, BC=26BC = 26.

Final Answer:

BC=26BC = 26

Would you like more details or have any further questions?

Here are five related questions for further practice:

  1. If AC=10x15AC = 10x - 15 and AB=4x+3AB = 4x + 3, how do you find BCBC?
  2. What happens to the length of BCBC if BB is no longer the midpoint of ACAC?
  3. How would the equation change if ABAB and BCBC were in a ratio of 2:1?
  4. If ACAC was expressed as a quadratic function, how would you solve for the length of BCBC?
  5. What if ABAB was a function of another variable, say yy? How would you approach the solution?

Tip: Always check midpoint problems by confirming that AB=BCAB = BC, which ensures your equation setup is correct.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Midpoint

Formulas

AC = AB + BC
Since B is the midpoint: AB = BC

Theorems

Midpoint Theorem

Suitable Grade Level

Grades 8-10