In the given image, the right triangle $\triangle ABC$ has squares drawn on two of its sides:

1. One square on side $AB$ has an area of $50 \, \text{sq. cm}$.
2. The other square on side $AC$ has an area of $18 \, \text{sq. cm}$.

Step 1: Calculate the lengths of sides $AB$ and $AC$

The area of a square is given by $\text{Area} = \text{side}^2$. So, we can find the lengths of sides $AB$ and $AC$ by taking the square roots of their areas.

• For side $AB$: $\text{Length of } AB = \sqrt{50} \approx 7.07 \, \text{cm}$

• For side $AC$: $\text{Length of } AC = \sqrt{18} \approx 4.24 \, \text{cm}$

Step 2: Use the Pythagorean Theorem to find side $BC$

Since this is a right triangle, the Pythagorean theorem applies: $AB^2 = AC^2 + BC^2$ We need to find $BC$, so we rearrange the equation: $BC^2 = AB^2 - AC^2$ Substitute the values: $BC^2 = 50 - 18 = 32$ Thus, the length of $BC$ is: $BC = \sqrt{32} \approx 5.66 \, \text{cm}$

Step 3: Calculate the perimeter of the triangle

The perimeter of the triangle is the sum of the lengths of all three sides: $\text{Perimeter} = AB + AC + BC = 7.07 + 4.24 + 5.66 = 16.97 \, \text{cm}$

Final Answer:

• The length of the third side $BC$ is approximately $5.66 \, \text{cm}$.
• The perimeter of the triangle is approximately $16.97 \, \text{cm}$.

Would you like more details on any of these calculations?

Here are some related questions you could explore:

1. How does the Pythagorean theorem apply to different geometric shapes?
2. Can you find the area of $\triangle ABC$ using Heron's formula with the calculated side lengths?
3. How would the perimeter change if the areas of the squares were different?
4. What other geometric properties of right triangles are useful in practical applications?
5. What is the relationship between the areas of the squares and the triangle's sides?

Tip: Always check if a triangle is a right triangle by verifying if the Pythagorean theorem holds true for its side lengths!