To find the perimeter of the right-angled triangle, we'll start by labeling the sides. Let the shortest side be [tex]\(a\)[/tex] and the next shortest side be [tex]\(b\)[/tex]. We are given the following information:
1. The area of the triangle is 54 cm².
2. The shortest side [tex]\(a\)[/tex] is 3 cm less than the next shortest side [tex]\(b\)[/tex].
Thus, we can write:
[tex]\[ b = a + 3 \][/tex]
The area [tex]\(A\)[/tex] of a right-angled triangle can be calculated using the formula:
[tex]\[ A = \frac{1}{2} \times a \times b \][/tex]
Substituting the given area and the relationship between [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ 54 = \frac{1}{2} \times a \times (a + 3) \][/tex]
Multiplying both sides by 2 to eliminate the fraction:
[tex]\[ 108 = a \times (a + 3) \][/tex]
This simplifies to the quadratic equation:
[tex]\[ a^2 + 3a - 108 = 0 \][/tex]
We solve this quadratic equation for [tex]\(a\)[/tex]. The positive root of the quadratic equation is [tex]\(a = 9\)[/tex] (since a side length cannot be negative).
Next, calculate the length of the next shortest side [tex]\(b\)[/tex]:
[tex]\[ b = a + 3 = 9 + 3 = 12 \, \text{cm} \][/tex]
Now, we find the hypotenuse [tex]\(c\)[/tex] of the right-angled triangle using the Pythagorean theorem:
[tex]\[ c = \sqrt{a^2 + b^2} \][/tex]
[tex]\[ c = \sqrt{9^2 + 12^2} \][/tex]
[tex]\[ c = \sqrt{81 + 144} \][/tex]
[tex]\[ c = \sqrt{225} \][/tex]
[tex]\[ c = 15 \, \text{cm} \][/tex]
Finally, we can determine the perimeter of the triangle by summing the lengths of all sides:
[tex]\[ \text{Perimeter} = a + b + c \][/tex]
[tex]\[ \text{Perimeter} = 9 + 12 + 15 \][/tex]
[tex]\[ \text{Perimeter} = 36 \, \text{cm} \][/tex]
Thus, the shortest side is 9 cm, the next shortest side is 12 cm, the hypotenuse is 15 cm, and the perimeter of the triangle is 36 cm.
