Question

the shortest leg of a triangle is 7 meters shorter than the other leg. the hypotenuse of this triangle is 1 meters. what are the lengths of the two legs of this triangle? the shortest leg is meters long. the other leg is meters long.

Explanation:

Step1: Define variables

Let the length of the other leg be \( x \) meters. Then the length of the shortest leg is \( x - 7 \) meters. Since it's a right triangle (we can assume it's a right triangle as we are dealing with legs and hypotenuse), we can use the Pythagorean theorem. The hypotenuse is given (assuming it's 13 meters, as the text seems to have a typo, likely 13). The Pythagorean theorem states that for a right triangle with legs \( a \), \( b \) and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). Here, \( a=x - 7 \), \( b = x \) and \( c=13 \). So we have the equation \((x - 7)^{2}+x^{2}=13^{2}\).

Step2: Expand and simplify the equation

First, expand \((x - 7)^{2}\): \((x - 7)^{2}=x^{2}-14x + 49\). Then the equation becomes \(x^{2}-14x + 49+x^{2}=169\). Combine like terms: \(2x^{2}-14x+49 - 169 = 0\), which simplifies to \(2x^{2}-14x - 120=0\). Divide the entire equation by 2: \(x^{2}-7x - 60 = 0\).

Step3: Solve the quadratic equation

We can factor the quadratic equation \(x^{2}-7x - 60 = 0\). We need two numbers that multiply to - 60 and add up to - 7. The numbers are - 12 and 5. So the equation factors as \((x - 12)(x + 5)=0\). Setting each factor equal to zero gives \(x - 12 = 0\) or \(x+5 = 0\). Solving these, we get \(x = 12\) or \(x=-5\). Since length can't be negative, we discard \(x=-5\).

Step4: Find the length of the shortest leg

If the other leg \(x = 12\) meters, then the shortest leg is \(x - 7=12 - 7 = 5\) meters.

Answer:

The shortest leg is \(\boldsymbol{5}\) meters long.
The other leg is \(\boldsymbol{12}\) meters long.