n a right-angled triangle, the lengths of two shorter sides are 3 cm and 4 cm. Can you find the area of thetriangle using both the Pythagorean theorem and Heron's formula? Explain your approach.

Sure, I can help with that.

First, let's find the length of the hypotenuse using the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, if we let c be the hypotenuse, a and b be the other two sides, we have:

c^2 = a^2 + b^2 c^2 = 3^2 + 4^2 c^2 = 9 + 16 c^2 = 25 c = sqrt(25) c = 5 cm

So, the length of the hypotenuse is 5 cm.

Now, let's find the area of the triangle. The area (A) of a right-angled triangle can be found using the formula:

A = 1/2 * base * height A = 1/2 * 3 cm * 4 cm A = 1/2 * 12 cm^2 A = 6 cm^2

So, the area of the triangle is 6 square cm.

Now, let's use Heron's formula. Heron's formula is used to find the area of a triangle when we know the lengths of all three sides. The formula is:

A = sqrt[s(s - a)(s - b)(s - c)]

where s is the semi-perimeter of the triangle, calculated as (a + b + c) / 2.

So, let's calculate s:

s = (3 cm + 4 cm + 5 cm) / 2 s = 12 cm / 2 s = 6 cm

Now, we can find the area:

A = sqrt[6 cm(6 cm - 3 cm)(6 cm - 4 cm)(6 cm - 5 cm)] A = sqrt[6 cm * 3 cm * 2 cm * 1 cm] A = sqrt[36 cm^2] A = 6 cm^2

So, the area of the triangle is also 6 square cm using Heron's formula. This confirms our earlier calculation.