An isosceles triangle has two sides of equal length. The third side is 5 less than twice the length of one of the other sides. If the perimeter of the triangle is 23 cm, what is the length of the third side The third side is described in relation to one of the equal sides, so let x = the length of one of the equal sides.
The Correct Answer and Explanation is:
Let xxx be the length of one of the equal sides of the isosceles triangle.
Since it’s an isosceles triangle, two sides have the same length.
The third side is 5 less than twice the length of one of the equal sides. This means the third side is 2x−52x – 52x−5.
The perimeter of a triangle is the sum of all its sides.
So, the total perimeter is: x+x+(2x−5)=23x + x + (2x – 5) = 23x+x+(2x−5)=23
Now simplify the left-hand side: 2x+(2x−5)=232x + (2x – 5) = 232x+(2x−5)=23 4x−5=234x – 5 = 234x−5=23
Now solve for xxx: 4x=23+54x = 23 + 54x=23+5 4x=284x = 284x=28 x=7x = 7x=7
So each of the equal sides is 7 cm long.
The third side is: 2x−5=2(7)−5=14−5=92x – 5 = 2(7) – 5 = 14 – 5 = 92x−5=2(7)−5=14−5=9
Answer: The third side is 9 cm long.
Explanation
To find the length of the third side of the isosceles triangle, we start by using a variable. Let the length of one of the equal sides be xxx. Since it is an isosceles triangle, the second equal side also has a length of xxx. The problem tells us that the third side is 5 centimeters less than twice the length of one of these equal sides. In mathematical terms, that is represented as 2x−52x – 52x−5.
Next, we apply the perimeter formula for a triangle. The perimeter is the sum of the lengths of all three sides. Therefore, the perimeter is x+x+(2x−5)x + x + (2x – 5)x+x+(2x−5). This simplifies to 4x−54x – 54x−5. According to the problem, the perimeter is 23 centimeters. So, we set up the equation: 4x−5=234x – 5 = 234x−5=23
To solve this equation, we first add 5 to both sides, resulting in 4x=284x = 284x=28. Then, divide both sides by 4 to isolate xxx, giving x=7x = 7x=7. This means each of the equal sides is 7 centimeters.
Now, substitute this value back into the expression for the third side: 2x−5=14−5=92x – 5 = 14 – 5 = 92x−5=14−5=9. So, the third side is 9 centimeters.
Finally, we verify our answer by checking the perimeter: 7+7+9=237 + 7 + 9 = 237+7+9=23, which matches the given perimeter. The answer is correct.
