Problem Analysis:

You provided a problem that asks for the area and perimeter of $\Delta ABC$. Observing the diagram:

1. $AB = 4$ (horizontal length)
2. $BC = 5$ (vertical length, shown as the height from $C$ to $AB$).
3. $AC$ is the hypotenuse of the triangle (not given directly but can be calculated using the Pythagorean theorem).

Solution Steps:

1. Calculate the hypotenuse $AC$ using the Pythagorean theorem:

$AC = \sqrt{AB^2 + BC^2}$ $AC = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41}$

2. Compute the Area of $\Delta ABC$:

The area of a right triangle is: $\text{Area} = \frac{1}{2} \cdot \text{Base} \cdot \text{Height}$ Here, the base is $AB = 4$, and the height is $BC = 5$: $\text{Area} = \frac{1}{2} \cdot 4 \cdot 5 = 10 \, \text{square units.}$

3. Compute the Perimeter of $\Delta ABC$:

The perimeter is the sum of all three sides: $\text{Perimeter} = AB + BC + AC$ $\text{Perimeter} = 4 + 5 + \sqrt{41} \approx 4 + 5 + 6.4 = 15.4 \, \text{units.}$

Final Answer:

• Area: $10 \, \text{square units}$
• Perimeter: $4 + 5 + \sqrt{41} \approx 15.4 \, \text{units.}$

Would you like a detailed step-by-step explanation of any part?

Follow-up Questions:

1. How can the Pythagorean theorem be applied to verify a right triangle?
2. What are some alternative methods to find the area of a triangle without knowing the height explicitly?
3. How would the calculations change if the triangle was not a right triangle?
4. Can the area of this triangle be calculated using the coordinates of the vertices?
5. What would be the implications of scaling all sides of the triangle by a constant factor?

Tip:

To verify if a triangle is a right triangle, check if the square of the longest side equals the sum of the squares of the other two sides!