Respuesta :
Answer:
let x be the shortest side
let y be the other side
let z be the hypotenuse
.: y = 7 + x
z = 3x - 2
Answer:
The lengths of the sides of the triangle are 5 cm, 12 cm, and 13 cm, as given in the answer.
Step By Step Explanation:
Let's denote the lengths of the sides of the right-angled triangle as follows:
- Let a be the length of the shortest side.
- Let b be the length of the other side (which is 7 cm longer than the shortest side).
- Let c be the length of the hypotenuse (which is two centimeters shorter than three times the length of the shortest side).
Now, we can set up equations:
The Pythagorean Theorem: In a right-angled triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):
[tex]c^2 = a^2+b^2[/tex]
The given relationship between the hypotenuse and the shortest side: [tex]c = 3a -2[/tex]
The other side is 7 cm longer than the shortest side:
[tex]b= a+7[/tex]
Now, we can use these equations to solve for the lengths of a, b, and c.
Substitute the expression for cc from equation (2) into equation (1):
[tex](3a-2)^2 = a^2 + (a+7)^2[/tex]
Now, solve this equation to find the value of a.
[tex]9a^2 -12a+4 = a^2+a^2+14a+ 49[/tex]
[tex]9a^2-12a+4 = 2a^2+14a+49[/tex]
[tex]7a^2-26a-45=0[/tex]
Now, factor the quadratic equation:
[tex](7a+9)(a-5)=0[/tex]
So,
[tex]a= -\frac{9}{7}[/tex] or [tex]a=5[/tex]
Since the length cannot be negative, we discard the first solution.
So, the length of the shortest side (a) is 5 cm. Now, use this value to find b and c:
[tex]b=a+7= 5+7=12cm[/tex]
[tex]c=3a-2= 3(5)-2=13cm[/tex]
