An isosceles triangle has legs that are each x inches long a base that is y inches long. The perimeter of this triangle is 38 inches. The base is 8 inches shorter than the length of a leg.
2x + y = 38 -----> y = 38-2x
the base is 8 inches shorter than the length of a leg
y = x-8
x-8 = 38-2x
3x = 46
x = 46/3 inches
then y = 38 - 92/3 = 22/3 inches
check:
perimeter = 2x+y = 92/3+22/3 = 38
difference between leg and base
= 46/3 - 22/3 = 8
Let's break down the information given step by step:
1. The triangle is isosceles, meaning it has two equal sides (legs) and one base.
2. The length of each leg is x inches.
3. The length of the base is y inches.
4. The perimeter of the triangle is 38 inches, so the sum of all three sides is 38 inches.
5. The base is 8 inches shorter than the length of a leg.
Now, let's use this information to find the values of x and y.
Step 1: Perimeter equation
The perimeter of a triangle is the sum of all three sides, so we can write the equation as:
x + x + y = 38
Step 2: Base length equation
The base is 8 inches shorter than the length of a leg, so we can write the equation as:
y = x - 8
Step 3: Substitute the second equation into the first equation
Substituting the value of y from equation 2 into equation 1, we get:
x + x + (x - 8) = 38
3x - 8 = 38
Step 4: Solve for x
Adding 8 to both sides of the equation:
3x = 46
Dividing both sides by 3:
x = 46 / 3
Step 5: Calculate the value of x
x ≈ 15.33
Step 6: Calculate the value of y
Substituting the value of x into equation 2:
y = 15.33 - 8
y ≈ 7.33
Therefore, in this isosceles triangle, the length of each leg is approximately 15.33 inches, and the length of the base is approximately 7.33 inches.
To solve this problem, we need to use the information given to find the values of x and y, which represent the lengths of the legs and the base of the triangle, respectively. Let's break it down step by step.
1. Let's first represent the length of each leg as x inches.
So, the length of one leg is x inches, and since it is an isosceles triangle, the other leg will also be x inches.
2. As given, the base is 8 inches shorter than the length of a leg.
So, the length of the base can be represented as (x - 8) inches.
3. The perimeter of a triangle is the sum of the lengths of its sides.
In our case, the perimeter of the triangle is 38 inches, and it consists of two legs and one base.
Therefore, the equation for the perimeter can be written as: x + x + (x - 8) = 38.
4. Simplifying the equation, we have: 2x + (x - 8) = 38.
Combining like terms, we get: 3x - 8 = 38.
Adding 8 to both sides of the equation, we have: 3x = 46.
Dividing both sides by 3, we find: x = 46/3.
5. Therefore, the length of each leg (x) is 46/3 inches.
6. Finally, we can calculate the length of the base (y) by substituting the value of x into our expression for the base: y = x - 8.
y = (46/3) - 8.
By following these steps, we have determined the lengths of the legs and the base of the isosceles triangle.