the cross- section of a prism is a right angle triangle 13cm by 12cm by 5cm,

Let's analyze the problem involving the prism with a cross-section in the shape of a right-angled triangle. The dimensions given are 13 cm, 12 cm, and 5 cm.

To understand the problem fully, we need to verify if the given sides indeed form a right-angled triangle, and then we can discuss further properties such as area or volume if required.

First, let's verify if these sides form a right-angled triangle using the Pythagorean theorem:

\[ a^2 + b^2 = c^2 \]

Here, the largest side (13 cm) should be the hypotenuse if it's a right-angled triangle. Let's check:

\[ 12^2 + 5^2 = 13^2 \]
\[ 144 + 25 = 169 \]
\[ 169 = 169 \]

Since the equation holds true, these sides indeed form a right-angled triangle with the hypotenuse 13 cm.

Now, we can find the area of the cross-sectional right-angled triangle:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
\[ \text{Area} = \frac{1}{2} \times 12 \times 5 \]
\[ \text{Area} = \frac{1}{2} \times 60 \]
\[ \text{Area} = 30 \, \text{cm}^2 \]

If you need further calculations like the volume of the prism, you would need the length of the prism. For now, we have confirmed the sides form a right-angled triangle and found the area of the cross-section.

Would you like more details or have any other questions? Here are some more problems you might find interesting:

1. Calculate the volume of the prism if the length is 20 cm.
2. Determine the perimeter of the triangular cross-section.
3. Find the height of the prism if its volume is 600 cm³.
4. Calculate the surface area of the prism with a given length of 10 cm.
5. If another prism has a cross-section of an equilateral triangle with a side of 6 cm, compare its area with the given prism.

**Tip:** Always verify the triangle's properties using the Pythagorean theorem when given three sides to ensure it forms a right-angled triangle.

Learn how to calculate the area of a right-angled triangle prism cross-section with dimensions 13 cm, 12 cm, and 5 cm. Verify triangle properties using the Pythagorean theorem and compute the area. Suitable for students in Grades 7-9.

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