if the sides of a triangle are 8 cm, 10 cm, and 12 cm then find the length of altitude to the smaller side.
To find the length of the altitude to the smaller side of a triangle, we can use the formula for the area of a triangle. The area of a triangle can be calculated using the formula:
Area = (1/2) * base * height
In this case, the smaller side of the triangle can be considered as the base, and we need to find the length of the altitude, which will be the height. Let's call the smaller side of the triangle as side a and the altitude as h.
Since we have the lengths of all the sides of the triangle, we can use Heron's formula to calculate the area. Heron's formula states that the area of a triangle with sides a, b, and c is given by:
Area = sqrt(s * (s - a) * (s - b) * (s - c))
Where s is the semi-perimeter of the triangle, which is defined as:
s = (a + b + c) / 2
In this case, a = 8 cm, b = 10 cm, and c = 12 cm. So, we can calculate the semi-perimeter as follows:
s = (8 + 10 + 12) / 2
s = 30 / 2
s = 15 cm
Now, we can use Heron's formula to calculate the area of the triangle:
Area = sqrt(15 * (15 - 8) * (15 - 10) * (15 - 12))
Area = sqrt(15 * 7 * 5 * 3)
Area = sqrt(1575)
Area ≈ 39.69 cm²
Now that we have the area of the triangle, we can use the formula for the area of a triangle to find the length of the altitude to the smaller side. Rearranging the formula, we get:
h = (2 * Area) / base
Since the smaller side is the base, we can substitute the values:
h = (2 * 39.69) / 8
h ≈ 4.96 cm
Therefore, the length of the altitude to the smaller side of the triangle is approximately 4.96 cm.