Question 1210464: Find AE
BE = 30, PE = 18, CP = 10
Found 3 solutions by CPhill, timofer, ikleyn:
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! The problem can be solved by applying the **Intersecting Chords Theorem** to the circle that passes through the four points $A, B, C, E$.
## 🎯 Finding AE using the Intersecting Chords Theorem
The Intersecting Chords Theorem states that if two chords, $AC$ and $BE$, intersect at a point $P$ inside a circle, then the product of the segments of one chord equals the product of the segments of the other chord.
In this case, we have two chords, $AC$ and $BE$, intersecting at $P$.
1. **Identify the Segments and Given Lengths:**
* Chord $AC$ is divided into segments $AP$ and $CP$.
* Chord $BE$ is divided into segments $BP$ and $PE$.
* We are given:
* $BE = 30$ (Length of the entire chord)
* $PE = 18$
* $CP = 10$
2. **Calculate the length of the segment BP:**
The segment $BP$ is the difference between the entire chord $BE$ and the segment $PE$.
$$BP = BE - PE$$
$$BP = 30 - 18 = 12$$
3. **Apply the Intersecting Chords Theorem:**
The theorem states:
$$AP \cdot CP = BP \cdot PE$$
4. **Solve for the unknown segment AP:**
Substitute the known values into the equation:
$$AP \cdot 10 = 12 \cdot 18$$
$$10 \cdot AP = 216$$
$$AP = \frac{216}{10} = 21.6$$
5. **Calculate the length of AE:**
The question asks for the length of $AE$. However, in a standard diagram where chords $AC$ and $BE$ intersect at $P$, $AE$ is a **non-chord segment** (it's a side of $\triangle APE$).
**Assuming the request meant the length of the chord AC, as it uses all given information:**
* If the request was for $AC$:
$$AC = AP + CP$$
$$AC = 21.6 + 10 = \mathbf{31.6}$$
**Assuming the points form a cyclic quadrilateral $ABCE$ and the question truly asks for $AE$:**
We would need more information (like an angle or the radius of the circle) to find the length of $AE$ using the Law of Cosines on $\triangle APE$. The given lengths $AP=21.6, PE=18$ are sides, but the angle $\angle APE$ is missing.
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## ⭐ Finding AE using Power of a Point Theorem (Alternative)
If the points $A, B, C, E$ form a cyclic quadrilateral and the lines $AC$ and $BE$ are extended to intersect **outside** the circle at a point $P$ (the **Secant-Secant Theorem**), the relationship is different:
$$PA \cdot PC = PE \cdot PB$$
Given the phrasing and the internal intersection formula, it is overwhelmingly likely that **$AC$ was the intended segment to find.**
**Based on the only calculation that utilizes all given numbers through a standard circle theorem, the answer for the chord length AC is 31.6.** If the diagram implies a different theorem (like Secant-Tangent or Secant-Secant), please provide the configuration.
**Final Answer for the calculated segment AP:**
$$AP = 21.6$$
**Final Answer for the full chord AC (likely intended):**
$$AC = 31.6$$
Without a diagram, and given the ambiguous request for $AE$ vs. the solvable segments $AP$ and $AC$, the most complete answer is to provide the length of $AP$ and the probable intended length of $AC$.
Since the question asks for $AE$ and $AE$ cannot be found with the given information, I will provide the two relevant segments calculated: $AP$ and the full chord $AC$.
**Length of segment $AP$ is $\mathbf{21.6}$.**
**Length of chord $AC$ is $\mathbf{31.6}$.**
Answer by timofer(155)  (Show Source):
Answer by ikleyn(53763)  (Show Source):
You can put this solution on YOUR website! .
Find AE
BE = 30, PE = 18, CP = 10
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This problem is FATALLY DEFECTIVE.
Please STOP posting gibberish to this forum.
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