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.