Question

practice - radical equations and applications
18 points possible answered: 11/18
question 18
the shortest leg of a triangle is 1 meters shorter than the other leg. the hypotenuse of this triangle is 5 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: Set up variables

Let the length of the shortest leg be \(x\) meters. Then the length of the other leg is \((x + 1)\) meters.

Step2: Apply the Pythagorean theorem

According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(a=x\), \(b=x + 1\), \(c = 5\)). So we have the equation \(x^{2}+(x + 1)^{2}=5^{2}\).
Expand \((x + 1)^{2}\): \(x^{2}+x^{2}+2x + 1=25\).
Combine like - terms: \(2x^{2}+2x+1 - 25=0\), which simplifies to \(2x^{2}+2x-24 = 0\).
Divide the entire equation by \(2\): \(x^{2}+x - 12=0\).

Step3: Solve the quadratic equation

Factor the quadratic equation \(x^{2}+x - 12=(x + 4)(x - 3)=0\).
Set each factor equal to zero: \(x+4 = 0\) gives \(x=-4\) (rejected since length cannot be negative), \(x - 3=0\) gives \(x = 3\).

Answer:

The shortest leg is \(3\) meters long.
The other leg is \(4\) meters long.