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).

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 $AB = 9 \text{ cm}$ and $CD = 10.5 \text{ cm}$. This indicates an inconsistency in the problem statement. If $AB = CD$ were true, then $AP \times PB$ would equal $CP \times PD$ and $AP + PB$ would equal $CP + PD$. These two conditions cannot be simultaneously satisfied with the given segment lengths.

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