Answer:
26 units
Explanation:
The perimeter of a triangle is the sum of the lengths of its three sides. So, to find the perimeter of this triangle, we need to add x, (x-4), and (x² - 2x - 5).
P = x + (x-4) + (x² - 2x - 5) P = x + x - 4 + x² - 2x - 5 P = x² + 2x - 9
This is the expression for the perimeter of the triangle in terms of x. To find the numerical value, we need to plug in a value of x that is greater than 4. For example, if x = 5, then
P = (5)² + 2(5) - 9 P = 25 + 10 - 9 P = 26
So, the perimeter of the triangle is 26 units when x = 5. You can try other values of x that are greater than 4 and see how the perimeter changes.
footnotes:
- The reason x has to be greater than 4 is because of the side length (x-4). If x was less than or equal to 4, then (x-4) would be zero or negative, which is not possible for a side length of a triangle. For example, if x = 4, then (x-4) = 0, and the triangle would have no width. If x = 3, then (x-4) = -1, and the triangle would have a negative side length, which makes no sense. So, x has to be greater than 4 to ensure that all three sides are positive and form a valid triangle.
- If x was a fraction or a decimal, it could still be greater than 4. For example, if x = 4.5, then (x-4) = 0.5, which is a positive side length. However, x cannot be too close to 4, because then the third side length (x² - 2x - 5) would become negative or zero. For example, if x = 4.1, then (x² - 2x - 5) = -0.19, which is not a valid side length. So, x has to be greater than 4 by a certain amount to make sure that all three sides are positive and form a valid triangle. P = (6.5)² + 2(6.5) - 9 P = 42.25 + 13 - 9 P = 46.25. So, the perimeter of the triangle is 46.25 units when x = 6.5.
•First, I used the formula for the perimeter of a triangle, which is the sum of the lengths of its three sides.
•Second, I substituted the given expressions for the side lengths in terms of x: x, (x-4), and (x² - 2x - 5)
•Third, I simplified the expression by combining like terms: x + x - 4 + x² - 2x - 5 = x² + 2x - 9.
•Fourth, I plugged in the given value of x: 6.5, and evaluated the expression using the order of operations: (6.5)² + 2(6.5) - 9 = 42.25 + 13 - 9 = 46.25.
•Fifth, I wrote the answer with the correct units: 46.25 units.