Question

The hypotenuse of a right triangle is 6 cm and one side is 2 cm longer than the other side. Find the length of each side to one decimal place.

Answer 1

Answer:

The two legs of the right triangle are approximately 3.1 cm and 5.1 cm and the hypotenuse is 6 cm.

Step-by-step explanation:

Let x be one leg of the right triangle.

Since the other leg is two centimeters longer, it can be represented by the expression:

$x + 2 \text{ cm}$

According to the Pythagorean Theorem, for a right triangle:

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

Where c is the hypotenuse and a and b are the two legs.

The hypotenuse is given to be 6 cm and the two legs are x and (x + 2). Hence:

$(x)^2 + (x+2)^2 = 6^2$

Solve for x. Simplify:

$x^2 + (x^2 + 4x + 4) = 36$

Simplify:

$2x^2 + 4x + 4 = 36$

Subtract:

$2x^2 + 4x - 32 =0$

Divide:

$x^2 + 2x - 16 = 0$

The equation is not factorable, so we can consider using the quadratic formula:

$\displaystyle x = \frac{-b\pm\sqrt{b^2 -4ac}}{2a}$

In this case, a = 1, b = 2, and c = -16. Substitute:

$\displaystyle x= \frac{-(2)\pm\sqrt{(2)^2-4(1)(-16)}}{2(1)}$

And evaluate:

$\displaystyle \begin{aligned}x &= \frac{-2\pm\sqrt{68}}{2} \\ \\ &= \frac{-2\pm\sqrt{4\cdot 17}}{2} \\ \\ &= \frac{-2\pm2\sqrt{17}}{2} \\ \\ &= -1 \pm \sqrt{17} \end{aligned}$

Hence, our two solutions are:

$\displaystyle x = -1 + \sqrt{17} \approx 3.1\text{ or } x = -1 - \sqrt{17} \approx -5.1$

Lengths cannot be negative, so we can ignore the second solution.

Hence, the value of x or the first side length is about 3.1 centimeters.

Since the other side length is two centimeters longer, the other side is about 5.1 centimeters.

In conclusion, the two legs of the right triangle are approximately 3.1 cm and 5.1 cm and the hypotenuse is 6 cm.