Question

the sides of a triangle have lengths 15, 20, 25. find the length of the shortest altitude of the triangles.

Answer 1

The shortest altitude of the right triangle with sides 15, 20, and 25 is found by using the area formula A = 1/2 * base * height twice; first to calculate the area with the legs as the base and height, and then to find the altitude from the area and the hypotenuse as the base. The length of this altitude is 12 units.

Step-by-step explanation:

To find the length of the shortest altitude of a triangle with sides of lengths 15, 20, and 25, we first recognize that this is a right triangle (Pythagorean triplet 3:4:5 scaled by 5). The shortest altitude of a right triangle is the perpendicular height from the hypotenuse to the opposite vertex. To find this altitude, we can use the formula for the area of a triangle, A = 1/2 * base * height, where the base is the hypotenuse in this case.

First, we calculate the area of the triangle using the two legs as the base and height:

1. A = 1/2 * 15 * 20
2. A = 150 square units

Now, we use the hypotenuse (25 units) as the base to express the height (shortest altitude), h.

1. 150 = 1/2 * 25 * h
2. h = 150 / (1/2 * 25)
3. h = 12 units

Therefore, the length of the shortest altitude of the triangle is 12 units.

Answer 2

To find the length of the shortest altitude of a triangle with side lengths 15, 20, and 25, we can use the formula A = 1/2 * base * height, along with Heron's formula to calculate the area. The length of the shortest altitude is 20 units.

Step-by-step explanation:

To find the length of the shortest altitude of a triangle, we can use the formula A = 1/2 * base * height, where A is the area of the triangle and base is the length of the side the altitude is drawn from. In this case, we can choose any side as the base. Let's choose the side with length 15 as the base:

Area (A) = 1/2 * 15 * height

Area can also be calculated using Heron's formula:

Area = sqrt(s * (s - a) * (s - b) * (s - c)), where a, b, and c are the lengths of the sides of the triangle, and s is the semi-perimeter (s = (a + b + c) / 2).

Using the given side lengths (15, 20, 25) in Heron's formula, we have:

Area = sqrt((15+20+25)/2 * ((15+20+25)/2 - 15) * ((15+20+25)/2 - 20) * ((15+20+25)/2 - 25))

Calculating the area gives us:

Area = sqrt(30 * 15 * 10 * 5) = sqrt(22500) = 150 units squared

Now, we can solve for the height (the altitude):

150 = 1/2 * 15 * height

height = 150 / (1/2 * 15) = 150 / 7.5 = 20 units

Therefore, the length of the shortest altitude of the triangle is 20 units.