The sides and classifications of a triangle are given below. The length of the longest side is the integer given. What value(s) of x make the triangle?
x,x, 8; right
Please provide an explanation as well.
Thank you.
apparently you want the legs equal, in which case
x^2+x^2 = 8^2
2x^2 = 64
x^2 = 32
x = √32 = 4√2
To determine what values of x make a triangle with sides x, x, and 8 where the triangle is right, we need to apply the Pythagorean theorem.
According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Therefore, for our triangle where the sides are x, x, and 8, we have:
x^2 + x^2 = 8^2
Simplifying the equation, we get:
2x^2 = 64
Dividing both sides of the equation by 2, we get:
x^2 = 32
To solve for x, we can take the square root of both sides of the equation:
√(x^2) = √32
Since we are looking for a positive value for x, we can ignore the negative square root. Hence, we have:
x = √32 ≈ 5.66
Therefore, the value of x that makes the triangle is approximately 5.66.
To determine the possible values of x that would make the triangle described, we need to apply the triangle inequality theorem. According to this theorem, for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
In this case, the given side lengths are x, x, and 8, with the longest side being 8. Therefore, we can set up the following inequality:
x + x > 8
Simplifying, we have:
2x > 8
Now, we can solve for x by dividing both sides of the inequality by 2:
x > 4
This means that x must be greater than 4 for the triangle to exist. Since there are no other constraints or limitations given, the range of valid values for x would be any real number greater than 4.
So, to answer your question, any value of x that is greater than 4 will make the triangle described with sides x, x, and 8, and it will also be a right triangle according to the given classification.