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To determine if triangle ABC is a right-angled triangle, we can use the Pythagorean theorem, which states that for a right-angled triangle with sides \(a\), \(b\), and hypotenuse \(c\):

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

Given:
- \( AB = 8 \, \text{cm} \)
- \( BC = 5 \, \text{cm} \)
- Perimeter of the triangle \( = 22 \, \text{cm} \)

### Step 1: Determine the length of side \( AC \)

The perimeter of the triangle is the sum of all sides:
\[
AB + BC + AC = 22 \, \text{cm}
\]
Substituting the known values:
\[
8 \, \text{cm} + 5 \, \text{cm} + AC = 22 \, \text{cm}
\]
\[
AC = 22 \, \text{cm} - 13 \, \text{cm} = 9 \, \text{cm}
\]

So, \( AC = 9 \, \text{cm} \).

### Step 2: Apply the Pythagorean theorem

We now check if \( AC^2 = AB^2 + BC^2 \) or any other combination of sides forms a right-angled triangle.

Calculate the squares of the sides:
\[
AB^2 = 8^2 = 64
\]
\[
BC^2 = 5^2 = 25
\]
\[
AC^2 = 9^2 = 81
\]

Now, check if the sum of the squares of any two sides equals the square of the third side:
\[
AB^2 + BC^2 = 64 + 25 = 89
\]
\[
AC^2 = 81
\]

Since \( 89 \neq 81 \), the triangle ABC is **not** a right-angled triangle.

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Do you want more details or have any questions? 

Here are some related questions:
1. What are the properties of a right-angled triangle?
2. Can a triangle with sides 8 cm, 5 cm, and 9 cm be isosceles?
3. What is the formula for the perimeter of a triangle?
4. How do you determine the hypotenuse in a right-angled triangle?
5. What is the converse of the Pythagorean theorem?

**Tip:** Always verify the triangle's side lengths and ensure the conditions meet the criteria when applying the Pythagorean theorem.

Explore how to determine if triangle ABC with sides AB = 8 cm, BC = 5 cm, and AC = 9 cm is a right-angled triangle using the Pythagorean theorem and perimeter calculation. This problem involves basic geometry concepts and is suitable for Grades 7-9 students.

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