Find the perimeter and area of a right triangle if one leg measures 9 cm and the other leg measures 40 cm. Students must show all work step by step to receive full credit for the problem. Include correct units with your solution.

"...of a right triangle" is a trigger to the use of Pythagoras theorem to find the third side given the lengths of two legs.

So the length of the third side, S, is:
S=sqrt(9^2+40^2)
which should come out as an integer.
Adding the lengths of all three sides will give the required perimeter.

The are of the triangle is the product of the two legs divided by two.

To find the perimeter of a right triangle, you need to add up the lengths of all three sides. In this case, we are given the lengths of two legs.

1. Start by labeling the given sides:
- Leg 1 = 9 cm
- Leg 2 = 40 cm

2. To find the length of the hypotenuse (the third side), use the Pythagorean theorem. The theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let's represent the hypotenuse as 'c'.

Using the formula: c² = a² + b² (where a and b are the legs), we can calculate the length of the hypotenuse.

c² = 9² + 40²
c² = 81 + 1600
c² = 1681

Taking the square root of both sides: c = √1681 = 41 cm (rounded to the nearest whole number)

3. Now that we have all three side lengths, we can calculate the perimeter of the triangle by adding the lengths of all three sides.

Perimeter = Leg 1 + Leg 2 + Hypotenuse
= 9 cm + 40 cm + 41 cm
= 90 cm

Therefore, the perimeter of the right triangle is 90 cm.

4. To calculate the area of a right triangle, we can use the formula: Area = 0.5 * base * height.
In a right triangle, the legs serve as the base and height, so we can use either leg.

Let's use Leg 1 = 9 cm as the base, and Leg 2 = 40 cm as the height.

Area = 0.5 * 9 cm * 40 cm
= 0.5 * 360 cm²
= 180 cm²

Therefore, the area of the right triangle is 180 cm².

To summarize:
- The perimeter of the right triangle is 90 cm.
- The area of the right triangle is 180 cm².