Question
The length of the longer leg of a right triangle is 4ft longer than the length of the shorter leg x. The hypotenuse is 4ft shorter than twice the length of the shorter leg.
Write an equation in terms of x that represents the given relationship.
Write an equation in terms of x that represents the given relationship.
Answers
GPT-4o mini
Let's define the lengths of the sides of the right triangle based on the information provided:
- Let \( x \) be the length of the shorter leg.
- According to the problem, the length of the longer leg is \( x + 4 \) (4 feet longer than the shorter leg).
- The hypotenuse is given as \( 2x - 4 \) (4 feet shorter than twice the length of the shorter leg).
Now we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two legs. This can be expressed as:
\[
(\text{hypotenuse})^2 = (\text{shorter leg})^2 + (\text{longer leg})^2
\]
Substituting the expressions we defined earlier:
\[
(2x - 4)^2 = x^2 + (x + 4)^2
\]
Now, we can expand both sides:
1. Expand the left side:
\[
(2x - 4)^2 = 4x^2 - 16x + 16
\]
2. Expand the right side:
\[
(x + 4)^2 = x^2 + 8x + 16
\]
Thus, the right side becomes:
\[
x^2 + (x^2 + 8x + 16) = 2x^2 + 8x + 16
\]
Now we can put it all together into the equation:
\[
4x^2 - 16x + 16 = 2x^2 + 8x + 16
\]
Next, we can simplify this equation. First, subtract \( 2x^2 + 8x + 16 \) from both sides:
\[
4x^2 - 16x + 16 - 2x^2 - 8x - 16 = 0
\]
This simplifies to:
\[
2x^2 - 24x = 0
\]
Now we can factor this equation:
\[
2x(x - 12) = 0
\]
Setting each factor to zero gives us the solutions:
1. \( 2x = 0 \) → \( x = 0 \) (not a valid length for the triangle)
2. \( x - 12 = 0 \) → \( x = 12 \)
Thus, the equation that represents the relationship you asked for, before simplification, is:
\[
(2x - 4)^2 = x^2 + (x + 4)^2
\]
And, after simplifying it leads to the expression \( 2x(x - 12) = 0 \).
- Let \( x \) be the length of the shorter leg.
- According to the problem, the length of the longer leg is \( x + 4 \) (4 feet longer than the shorter leg).
- The hypotenuse is given as \( 2x - 4 \) (4 feet shorter than twice the length of the shorter leg).
Now we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two legs. This can be expressed as:
\[
(\text{hypotenuse})^2 = (\text{shorter leg})^2 + (\text{longer leg})^2
\]
Substituting the expressions we defined earlier:
\[
(2x - 4)^2 = x^2 + (x + 4)^2
\]
Now, we can expand both sides:
1. Expand the left side:
\[
(2x - 4)^2 = 4x^2 - 16x + 16
\]
2. Expand the right side:
\[
(x + 4)^2 = x^2 + 8x + 16
\]
Thus, the right side becomes:
\[
x^2 + (x^2 + 8x + 16) = 2x^2 + 8x + 16
\]
Now we can put it all together into the equation:
\[
4x^2 - 16x + 16 = 2x^2 + 8x + 16
\]
Next, we can simplify this equation. First, subtract \( 2x^2 + 8x + 16 \) from both sides:
\[
4x^2 - 16x + 16 - 2x^2 - 8x - 16 = 0
\]
This simplifies to:
\[
2x^2 - 24x = 0
\]
Now we can factor this equation:
\[
2x(x - 12) = 0
\]
Setting each factor to zero gives us the solutions:
1. \( 2x = 0 \) → \( x = 0 \) (not a valid length for the triangle)
2. \( x - 12 = 0 \) → \( x = 12 \)
Thus, the equation that represents the relationship you asked for, before simplification, is:
\[
(2x - 4)^2 = x^2 + (x + 4)^2
\]
And, after simplifying it leads to the expression \( 2x(x - 12) = 0 \).