We are given a circle $P$ with chords $AB$ and $CD$ intersecting at point $E$. We are also given the following measurements:

• $AD = AE = 3$
• $ED = 4$
• $BC = 5$

Our goal is to find the area of triangle $EBC$.

Step 1: Use the Power of a Point Theorem

The Power of a Point Theorem states that for two intersecting chords, the products of the segments of each chord are equal. Specifically, for the chords $AB$ and $CD$ intersecting at point $E$, we have the following relation:

$AE \times EB = CE \times ED$

We are given $AE = 3$, $AD = AE = 3$, and $ED = 4$. We do not yet know the lengths of $EB$ and $CE$, but we can express them in terms of each other using the Power of a Point.

Let $EB = x$ and $CE = y$. Then from the Power of a Point Theorem, we have:

$3 \times x = y \times 4$

Simplifying this equation:

$3x = 4y$ $x = \frac{4}{3}y \quad \text{(Equation 1)}$

Step 2: Apply the Length of Chord $BC$

We are also given the length of chord $BC$, which is $5$. Since $BC = BE + EC$, we have:

$BE + EC = 5$ $x + y = 5 \quad \text{(Equation 2)}$

Step 3: Solve the System of Equations

Now, we solve the system of equations (Equation 1 and Equation 2):

1. $x = \frac{4}{3}y$
2. $x + y = 5$

Substitute $x = \frac{4}{3}y$ into the second equation:

$\frac{4}{3}y + y = 5$

Combine the terms on the left-hand side:

$\frac{4}{3}y + \frac{3}{3}y = 5$ $\frac{7}{3}y = 5$

Multiply both sides by 3 to eliminate the fraction:

$7y = 15$ $y = \frac{15}{7}$

Now substitute $y = \frac{15}{7}$ back into $x = \frac{4}{3}y$:

$x = \frac{4}{3} \times \frac{15}{7} = \frac{60}{21} = \frac{20}{7}$

So, $x = \frac{20}{7}$ and $y = \frac{15}{7}$.

Step 4: Find the Area of Triangle $EBC$

The area of triangle $EBC$ can be found using the formula for the area of a triangle with base $BC$ and height $ED$. The base of triangle $EBC$ is $BC = 5$, and the height is the perpendicular distance from $E$ to the line $BC$, which is the length $ED = 4$.

The area of triangle $EBC$ is:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 5 \times 4 = 10$

Thus, the area of triangle $EBC$ is $\boxed{10}$.

Would you like further clarification or more details? Here are some related questions:

1. How does the Power of a Point Theorem apply to other geometric shapes?
2. Can the Power of a Point Theorem be used to find lengths in tangents to circles?
3. How do you calculate the area of a triangle when given the lengths of all sides?
4. What other methods can be used to calculate areas of triangles in coordinate geometry?
5. How do intersecting chords relate to the properties of inscribed angles?

Tip: When solving problems involving intersecting chords, always start by applying the Power of a Point Theorem, as it can simplify the relations between the segments of the chords.