Question

One leg of a right triangle is 1 cm less than twice the length of the other leg. The hypotenuse is 17 cm long. What are the lengths of the legs of the triangle?

Answer 1

Answer:

The length of shorter leg is 2.23 cm and the length of longer leg is 3.46 cm.

Step-by-step explanation:

The Pythagorean Theorem

If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

This relationship is represented by the formula:

                                                   $a^2+b^2=c^2$

Let x be length of shorter leg and 2x-1 be the length of longer leg.

By the Pythagorean Theorem,

$x^2+(2x-1)^2=17$

$5x^2-4x+1=17\\\\5x^2-4x-16=0$

Next, solve with the quadratic formula

$\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\\\x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$\mathrm{For\:}\quad a=5,\:b=-4,\:c=-16:\quad x_{1,\:2}=\frac{-\left(-4\right)\pm \sqrt{\left(-4\right)^2-4\cdot \:5\left(-16\right)}}{2\cdot \:5}$

Because the length can't be negative

$x=\frac{-\left(-4\right)+\sqrt{\left(-4\right)^2-4\cdot \:5\left(-16\right)}}{2\cdot \:5}= \frac{2\left(1+\sqrt{21}\right)}{5} \approx 2.23$

The length of shorter leg is 2.23 cm and the length of longer leg is $2\left(2.23\right)-1=3.46 \:cm$.