To solve for the area of triangle $\triangle ADE$, let's go through the problem using the given information:

Step 1: Analyze Given Information

• $AB = \sqrt{34}$ units.
• $AC = 3$ units.
• $CE = 21$ units.
• $\triangle CDE$ appears to be a right triangle at $C$.

Since $C$ is a right angle, we can apply the Pythagorean theorem in $\triangle CDE$ to find $DE$.

Step 2: Calculate $DE$ in $\triangle CDE$

In right triangle $\triangle CDE$: $DE = \sqrt{(AC)^2 + (CE)^2} = \sqrt{3^2 + 21^2} = \sqrt{9 + 441} = \sqrt{450} = 15\sqrt{2}$

Step 3: Calculate Area of $\triangle ADE$

The area of $\triangle ADE$ can be calculated as: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times DE \times AC$ Substitute $DE = 15\sqrt{2}$ and $AC = 3$: $\text{Area} = \frac{1}{2} \times 15\sqrt{2} \times 3 = \frac{1}{2} \times 45\sqrt{2} = 22.5\sqrt{2}$

Final Answer

The area of triangle $\triangle ADE$ is $22.5\sqrt{2}$ square units.

Would you like further details or have any questions?

Related Questions

1. How do you determine the base and height in a right triangle for area calculations?
2. What are the steps to simplify square roots in area calculations?
3. Can the Pythagorean theorem be applied to any triangle? Why or why not?
4. What other methods could be used to find the area of $\triangle ADE$?
5. How does the notation $\sqrt{34}$ impact exact answers versus approximate answers?

Tip

Always check for right angles in triangles; they often simplify calculations significantly with the Pythagorean theorem.