Find the area of triangle $ABC$.

[asy]
unitsize(1.5 cm);

pair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = foot(B,A,C),H = intersectionpoint(B--Q,A--P);
draw(A--B--C--cycle);
draw(A--P^^B--Q);
draw(rightanglemark(A,P,B,5));
draw(rightanglemark(B,Q,C,5));

label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("$P$",P,S); label("$Q$",Q,E); label("$H$",H,NW);
label("$6$", (A + H)/2, W, red);
label("$11$", (C + Q)/2, NE, red);
label("$4$", (A + Q)/2, NE, red);
[/asy]



We are given a triangle $ \triangle ABC $ with specific geometric properties. The key to solving the area involves using the heights provided in the diagram. Here's how we approach the solution:

### Step 1: Identify the base and height
From the diagram, we can interpret the following:

- The distance from point $ A $ to the line $ BC $, labeled as the perpendicular height, is 6 units.
- The base of the triangle is the length of side $ BC $, which is 11 units.

### Step 2: Apply the area formula for a triangle
The area $ A $ of a triangle is given by the formula:

\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]

Substituting the values we identified:

\[
A = \frac{1}{2} \times 11 \times 6 = \frac{1}{2} \times 66 = 33
\]

### Final Answer:
The area of $ \triangle ABC $ is $ \boxed{33} $ square units.

Would you like further details, or do you have any questions?

Here are 5 related questions to consider:
1. How would the area change if the base was doubled while keeping the height the same?
2. What is the formula for the area of a triangle if only the side lengths are known?
3. Can you find the perimeter of the triangle if the side lengths are given?
4. How does the Pythagorean theorem apply in a right-angled triangle?
5. How can the centroid of a triangle be found using its vertices?

**Tip**: The area of a triangle can also be found using trigonometry, especially when the angles between the sides are known!

Learn how to calculate the area of triangle ABC by applying the area formula for triangles. Given a base of 11 units and a height of 6 units, this problem provides a step-by-step solution to find the area of the triangle. Suitable for Grades 6-8 students.

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