Question
The length of the longer leg of a right triangle is 7ft longer than the length of the shorter leg x. The hypotenuse is 7ft 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
To represent the relationships in the right triangle using an equation in terms of the shorter leg \( x \), we can start by defining the lengths of the legs and the hypotenuse based on the information given:
1. Let the length of the shorter leg be \( x \).
2. The length of the longer leg is \( x + 7 \) (since it is 7 feet longer than the shorter leg).
3. The hypotenuse is \( 2x - 7 \) (since it is 7 feet shorter than twice the length of the shorter leg).
Using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse \( c \) is equal to the sum of the squares of the lengths of the other two legs \( a \) and \( b \):
\[
c^2 = a^2 + b^2
\]
In our case, we can substitute \( a \) with \( x \) (the shorter leg), \( b \) with \( x + 7 \) (the longer leg), and \( c \) with \( 2x - 7 \) (the hypotenuse). This gives us the following equation:
\[
(2x - 7)^2 = x^2 + (x + 7)^2
\]
Expanding both sides:
Left-hand side:
\[
(2x - 7)^2 = 4x^2 - 28x + 49
\]
Right-hand side:
\[
x^2 + (x + 7)^2 = x^2 + (x^2 + 14x + 49) = 2x^2 + 14x + 49
\]
Setting both sides equal to each other:
\[
4x^2 - 28x + 49 = 2x^2 + 14x + 49
\]
Now, we can simplify the equation. First, subtract \( 2x^2 + 14x + 49 \) from both sides:
\[
4x^2 - 28x + 49 - 2x^2 - 14x - 49 = 0
\]
This simplifies to:
\[
2x^2 - 42x = 0
\]
Factoring out \( 2x \):
\[
2x(x - 21) = 0
\]
Setting each factor to zero gives us:
1. \( 2x = 0 \) \(\Rightarrow x = 0\) (not applicable in this context since leg lengths cannot be zero)
2. \( x - 21 = 0\) \(\Rightarrow x = 21\)
Thus, the equation formed in terms of \( x \) representing the triangle's properties is:
\[
(2x - 7)^2 = x^2 + (x + 7)^2
\]
And the solution for \( x \) is \( x = 21 \) feet for the shorter leg.
1. Let the length of the shorter leg be \( x \).
2. The length of the longer leg is \( x + 7 \) (since it is 7 feet longer than the shorter leg).
3. The hypotenuse is \( 2x - 7 \) (since it is 7 feet shorter than twice the length of the shorter leg).
Using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse \( c \) is equal to the sum of the squares of the lengths of the other two legs \( a \) and \( b \):
\[
c^2 = a^2 + b^2
\]
In our case, we can substitute \( a \) with \( x \) (the shorter leg), \( b \) with \( x + 7 \) (the longer leg), and \( c \) with \( 2x - 7 \) (the hypotenuse). This gives us the following equation:
\[
(2x - 7)^2 = x^2 + (x + 7)^2
\]
Expanding both sides:
Left-hand side:
\[
(2x - 7)^2 = 4x^2 - 28x + 49
\]
Right-hand side:
\[
x^2 + (x + 7)^2 = x^2 + (x^2 + 14x + 49) = 2x^2 + 14x + 49
\]
Setting both sides equal to each other:
\[
4x^2 - 28x + 49 = 2x^2 + 14x + 49
\]
Now, we can simplify the equation. First, subtract \( 2x^2 + 14x + 49 \) from both sides:
\[
4x^2 - 28x + 49 - 2x^2 - 14x - 49 = 0
\]
This simplifies to:
\[
2x^2 - 42x = 0
\]
Factoring out \( 2x \):
\[
2x(x - 21) = 0
\]
Setting each factor to zero gives us:
1. \( 2x = 0 \) \(\Rightarrow x = 0\) (not applicable in this context since leg lengths cannot be zero)
2. \( x - 21 = 0\) \(\Rightarrow x = 21\)
Thus, the equation formed in terms of \( x \) representing the triangle's properties is:
\[
(2x - 7)^2 = x^2 + (x + 7)^2
\]
And the solution for \( x \) is \( x = 21 \) feet for the shorter leg.