A circle with centre O has two chords AB and CD such that AB = CD. The chords intersect at a point P inside the circle. It is given that:
* AP = 4 cm
* PB = 5 cm
* CP = 2.5 cm
Find:
1. The length of PD.
2. The length of each chord.
3. State the theorem that justifies your answer to part (2).
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Step-by-Step Solution
Step 1: Apply the Intersecting Chords Theorem
When two chords intersect inside a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. This is known as the Intersecting Chords Theorem.
Step 2: Substitute given values and solve for PD
We substitute the given lengths of the segments into the Intersecting Chords Theorem equation. Then, we solve for PD by dividing the product of AP and PB by CP.
Step 3: Calculate the lengths of the chords
The length of each chord is the sum of its segments. We add AP and PB to find the length of chord AB, and we add CP and PD to find the length of chord CD.
Step 4: Reconcile chord lengths and state theorem
The problem states that AB=CD. However, our calculations yield AB=9 cm and CD=10.5 cm. This indicates an inconsistency in the problem statement. If AB=CD were true, then AP×PB would equal CP×PD and AP + PB would equal CP + PD. These two conditions cannot be simultaneously satisfied with the given segment lengths.