Answer:
a) sides: 12 cm, 8 cm, 6 cm
b) perimeter: √455 ≈ 21.331 cm²
c) altitude: (√455)/6 ≈ 3.555 cm
Explanation:
The perimeter is the sum of side lengths. Heron's formula can be used to find the area from the side lengths, and the area formula can be used to find the altitude.
a) Perimeter
The perimeter is given as 26 cm, and the side lengths are listed as 6x, 4x, and 3x. The perimeter is the sum of side lengths, so we have ...
26 = 6x +4x +3x = 13x
x = 26/13 = 2 . . . . . . . . . divide by the coefficient of x
Then the side lengths are
6x = 6(2) = 12 . . . cm
4x = 4(2) = 8 . . . cm
3x = 3(2) = 6 . . . cm
b) Area
The area can be found using Heron's formula:
A = √(s(s -a)(s -b)(s -c)) . . . . . where s is half the perimeter; a, b, c are sides
A = √(13(13 -12)(13 -8)(13 -6)) = √(13(1)(5)(7)) = √455
The area of the triangle is √455 ≈ 21.331 square centimeters.
c) Altitude
The formula for the area of a triangle is ...
A = 1/2bh
where b is the length of one side, and h is the altitude to that side. We want the altitude to the longest side. Filling in the relevant values, we have ...
√455 = 1/2(12)h
h = (√455)/6 ≈ 3.555 . . . cm
The altitude to the longest side is (√455)/6 ≈ 3.555 centimeters.