The length of a line segment AB is 2 cm. It is divided into two parts at a point C such that AC 2= AB × CB. What is the length of CB?
3 – √5 cm
Understanding the Line Segment Division Problem
The question asks us to find the length of a segment CB, where a point C divides a line segment AB of a known length. We are given the total length of the line segment AB and a specific relationship between the lengths of the segments AC and CB, involving the total length AB.
Key Information Provided:
- The total length of the line segment AB is 2 cm.
- Point C lies on the line segment AB, dividing it into two parts: AC and CB.
- The relationship given is
AC2 = AB × CB.
We need to determine the length of CB based on this information.
Setting Up the Equation for CB Length
Let's denote the length of the line segment CB as \(x\) cm.
Since C is a point on the line segment AB, the length of AB is the sum of the lengths of AC and CB:
AB = AC + CB
We know AB = 2 cm and we've defined CB = \(x\) cm. So, we can find the length of AC:
\(2 = AC + x\)
\(AC = 2 - x\) cm.
Now, let's use the given condition: \(AC^2 = AB \times CB\).
Substitute the expressions for AC, AB, and CB into this equation:
\((2 - x)^2 = 2 \times x\)
Solving the Quadratic Equation for CB
We need to solve the equation \((2 - x)^2 = 2x\) for \(x\). Let's expand and rearrange the equation to form a standard quadratic equation (\(ax^2 + bx + c = 0\)).
- Expand the squared term:
\((2 - x)^2 = 2^2 - 2(2)(x) + x^2 = 4 - 4x + x^2\) - Set the expanded term equal to \(2x\):
\(4 - 4x + x^2 = 2x\) - Rearrange the equation to the standard quadratic form: Subtract \(2x\) from both sides.
\(x^2 - 4x - 2x + 4 = 0\)\(x^2 - 6x + 4 = 0\) - Solve the quadratic equation using the quadratic formula:
The formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
In our equation, \(a=1\), \(b=-6\), and \(c=4\).
\(x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(4)}}{2(1)}\)\(x = \frac{6 \pm \sqrt{36 - 16}}{2}\)\(x = \frac{6 \pm \sqrt{20}}{2}\) - Simplify the square root: \(\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}\).
\(x = \frac{6 \pm 2\sqrt{5}}{2}\) - Simplify the expression for x: Divide both terms in the numerator by 2.
\(x = \frac{2(3 \pm \sqrt{5})}{2}\)\(x = 3 \pm \sqrt{5}\)
This gives us two possible values for the length of CB (\(x\)):
- \(x_1 = 3 + \sqrt{5}\)
- \(x_2 = 3 - \sqrt{5}\)
Validating the Solution for CB
We need to determine which of the two possible values for \(x\) is the correct length for the segment CB.
Remember that CB (\(x\)) is a part of the line segment AB, which has a total length of 2 cm. Therefore, the length of CB must be less than 2 cm.
Let's evaluate the two possible solutions:
- Consider \(x_1 = 3 + \sqrt{5}\): We know that \(\sqrt{5}\) is approximately 2.236. So, \(x_1 \approx 3 + 2.236 = 5.236\) cm. Since 5.236 cm is greater than the total length of AB (2 cm), this solution is not physically possible for the length of CB.
- Consider \(x_2 = 3 - \sqrt{5}\): Using \(\sqrt{5} \approx 2.236\), we get \(x_2 \approx 3 - 2.236 = 0.764\) cm. This value is positive and less than 2 cm, which is a valid length for the segment CB.
We should also check if the corresponding length of AC is positive.
If \(x = 3 - \sqrt{5}\), then \(AC = 2 - x = 2 - (3 - \sqrt{5}) = 2 - 3 + \sqrt{5} = \sqrt{5} - 1\). Since \(\sqrt{5} \approx 2.236\), \(AC \approx 2.236 - 1 = 1.236\) cm, which is positive.
Thus, the only valid length for CB is \(3 - \sqrt{5}\) cm.
Final Answer Calculation Check
Let's verify the condition \(AC^2 = AB \times CB\) using the valid lengths:
- \(AB = 2\) cm
- \(CB = 3 - \sqrt{5}\) cm
- \(AC = \sqrt{5} - 1\) cm
Calculate \(AC^2\):
\(AC^2 = (\sqrt{5} - 1)^2 = (\sqrt{5})^2 - 2(\sqrt{5})(1) + 1^2 = 5 - 2\sqrt{5} + 1 = 6 - 2\sqrt{5}\)
Calculate \(AB \times CB\):
\(AB \times CB = 2 \times (3 - \sqrt{5}) = 6 - 2\sqrt{5}\)
Since \(AC^2 = 6 - 2\sqrt{5}\) and \(AB \times CB = 6 - 2\sqrt{5}\), the condition is satisfied.
Therefore, the length of CB is \(3 - \sqrt{5}\) cm.
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