Question

in the diagram, \\(ce = 12\\) units and \\(be = 5\\) units.
based on the given information, what is \\(ae\\)?

2 units
7 units
12 units
17 units

Explanation:

Identify the geometric properties from the diagram

The diagram shows a line \(l\) passing through points \(D\), \(B\), and \(E\).
The segment \(AC\) intersects line \(l\) at point \(B\).
The markings on the diagram indicate:

• \(AB \cong BC\) (indicated by the single tick marks on segments \(AB\) and \(BC\)).
• Line \(l\) is perpendicular to segment \(AC\) at point \(B\) (indicated by the right-angle symbol at intersection \(B\)).

Therefore, line \(l\) is the perpendicular bisector of segment \(AC\).

Apply the Perpendicular Bisector Theorem

Since line \(l\) is the perpendicular bisector of segment \(AC\), any point on line \(l\) is equidistant from the endpoints \(A\) and \(C\).
Point \(E\) lies on line \(l\).
Therefore:

$$ AE = CE $$

Calculate the length of AE

We are given:

$$ CE = 12\text{ units} $$

Using the equality from the previous step:

$$ AE = 12\text{ units} $$

Answer:

• 2 units
• 7 units
• 12 units (Correct answer)
• 17 units