The sides of a triangle have lengths of 3, 4, and 5. Find the area of the triangle.
right triangle
3^2 + 4^2 = 5^2
so
(1/2) base* altitude = (1/2)(4)(3) = 6
This is a right triangle.
A = 1/2bh
A = (1/2)(3)(4)
A = 6 square units
Given that you did not know that these three side lengths formed the smallest Pythagorean Triangle;
The area of a triangle can be derived from
1) A = bh/2 where b = the base and h = the altitude to the base.
2) A = ab(sinC)/2.
3) A = sqrt[s(s - a)(s - b)(s - c)] where a, b, and c are the three sides and s = the
semi-perimeter = (a + b + c)/2.
4) A = [4a^2b^2 - (c^2 - a^2 - b^2)^2]/4
5) A = rs where s is as defined above, and r is the radius of the incircle inscribed in
the triangle.
6) A = sqrt[r(ra)(rb)(rc)] in terms of the incircle and excircle radii.
7) A = s(s - c) for right triangles.
8) A = abc/4R
To find the area of a triangle, you can use the formula:
Area = (base * height) / 2
In this case, we can consider the side of length 5 as the base of the triangle. Now we need to find the height.
To find the height of a triangle, you can use the Pythagorean theorem. In a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this triangle with side lengths 3, 4, and 5, we can see that it is a right-angled triangle because 3^2 + 4^2 = 9 + 16 = 25, which is equal to 5^2.
Since the triangle is right-angled, the height of the triangle will be the length of the side that is perpendicular to the base. In this case, the height corresponds to the side of length 3 or 4.
Let's take the side of length 3 as the height of the triangle. Using the formula, we have:
Area = (base * height) / 2
= (5 * 3) / 2
= 15 / 2
= 7.5
Therefore, the area of the triangle is 7.5 square units.