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If AB=k+3, BC=2k and AC=5k−5, then for what value of 'k' does B lie on AC?
3
4
8
2
4
Since B lies on AC, simply sum the two segments AB and BC and equate the result to the total length AC: (k+3)+2k=5k−5.
AB = k + 3, BC = 2k, AC = 5k - 5. Point B lies on segment AC.
AB+BC=AC
Since B lies on AC, simply sum the two segments AB and BC and equate the result to the total length AC: (k+3)+2k=5k−5.
Students often assume AC is one of the smaller segments or try to set up an inequality instead of a collinearity equation (AB+BC=AC).
Set up the equation
If point B lies on line segment AC, then the sum of the lengths AB and BC must equal the total length of segment AC.
(k+3)+2k=5k−5
Simplify the expression
Combine the like terms on the left-hand side of the equation.
3k+3=5k−5
Solve for k
Isolate the variable k by moving 3k to the right and −5 to the left.
3+5=5k−3k⟹8=2k⟹k=4
B is correct because solving the equation AB+BC=AC for k yields k=4.
This concept of collinearity is foundational in coordinate geometry when checking if three points (x1,y1),(x2,y2),(x3,y3) lie on the same line.
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