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

Understand the Problem

The question is asking us to find the length of segment BC given that B is the midpoint of AC. We need to use the information that AC equals 8x - 20 to find the value of x and subsequently calculate BC.

Answer

The length of segment $ BC $ is $ 26 $.

Steps to Solve

Determine the Length of AC Given that segment ( AC = 8x - 20 ), this represents the total length of the segment between points A and C.
Use B as the Midpoint Since ( B ) is the midpoint of ( AC ), it means that ( AB = BC ). Therefore, we can express the length of ( BC ) as half of ( AC ):

$$ BC = \frac{AC}{2} $$

Substitute for AC Substituting the expression we have for ( AC ):

$$ BC = \frac{8x - 20}{2} $$

Simplify the Expression Now, simplify the expression for ( BC ):

$$ BC = 4x - 10 $$

Set Up the Equation Using ( AB ) We also know that ( AB = 3x - 1 ) (as indicated in the figure). Since ( AB = BC ), we can set the equations equal to each other:

$$ 3x - 1 = 4x - 10 $$

Solve for x Now, solve the equation for ( x ):

Rearranging gives us:

$$ 10 - 1 = 4x - 3x $$

$$ 9 = x $$

Calculate the Value of BC Substituting ( x = 9 ) back into the equation for ( BC ):

$$ BC = 4(9) - 10 $$

$$ BC = 36 - 10 $$

$$ BC = 26 $$

The length of segment ( BC ) is ( 26 ).

More Information

Since ( B ) is the midpoint, both segments ( AB ) and ( BC ) are equal. This allows us to set up the equation easily. Knowing ( AB ) is twice the size of ( BC ) simplifies the calculation.

Tips

Forgetting that the midpoint divides the segment into two equal parts can lead to incorrect calculations.
Incorrectly simplifying the expressions when solving for ( x ) can result in wrong values.

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