A triangle as a perimeter of 53 cm. The shortest side of the triangle is 2 cm more than 1/3 of the longest side and the other side is 5 cm less than the longer side. Find the dimensions.
let the longest side be x
shortest side = (1/3)x + 2
other side = x-5
x + (1/3)x + 2 + x - 5 = 53
times 3
3x + x + 6 + 3x - 15 = 159
7x = 168
x = 24
longest side = 24
shortest = 10
other = 19
check: 24+10+19 = 53
Let's denote the longest side of the triangle as x cm.
According to the given information, the shortest side is 2 cm more than 1/3 of the longest side. So, the shortest side can be expressed as (1/3)x + 2 cm.
The other side is 5 cm less than the longer side, so it can be expressed as x - 5 cm.
The perimeter of a triangle is calculated by adding the lengths of all three sides. So, the equation can be written as:
(1/3)x + 2 + x - 5 + x = 53
To solve this equation, first combine like terms:
(1/3)x + x + x + 2 - 5 = 53
(5/3)x - 3 = 53
Next, isolate the variable by adding 3 to both sides:
(5/3)x = 56
Then, multiply both sides by 3/5 to isolate x:
x = (56 * 3) / 5 = 33.6
Now that we have the longest side, we can find the other two sides.
The shortest side is (1/3)x + 2, so:
Shortest side = (1/3) * 33.6 + 2 = 11.2 + 2 = 13.2 cm
The other side is x - 5, so:
Other side = 33.6 - 5 = 28.6 cm
Therefore, the dimensions of the triangle are as follows:
Longest side: 33.6 cm
Shortest side: 13.2 cm
Other side: 28.6 cm
To find the dimensions of the triangle, we need to let x represent the length of the longest side.
Let's break down the given information step by step:
1. The shortest side is 2 cm more than 1/3 of the longest side:
Let's express this mathematically: (1/3)x + 2
2. The other side is 5 cm less than the longer side:
Mathematically, this can be represented as: x - 5
Now, we can set up an equation using the given information:
Perimeter of the triangle = Sum of all side lengths
53 cm = x + (1/3)x + 2 + x - 5
Simplifying the equation:
53 cm = (5/3)x - 3
To isolate x, the longest side, we can move all other terms to the right side of the equation:
53 + 3 = (5/3)x
56 = (5/3)x
To solve for x, we multiply both sides of the equation by the reciprocal of (5/3), which is (3/5):
(56) * (3/5) = x
x = 33.6 cm
Therefore, the longest side of the triangle measures 33.6 cm.
To find the lengths of the other two sides, we substitute this value back into the expressions we derived earlier:
Shortest side = (1/3)x + 2 = (1/3)(33.6) + 2 ≈ 13.2 cm
Other side = x - 5 = 33.6 - 5 = 28.6 cm
Hence, the dimensions of the triangle are approximately:
Longest side: 33.6 cm
Shortest side: 13.2 cm
Other side: 28.6 cm