Question

A right triangle has one leg twice as long as the other and the perimeter is 18. Find the three sides of the triangle. Draw a diagram.

Answer 1

To solve the problem, assign variables to the lengths of the two legs, set up an equation using the perimeter, rearrange and simplify, use the Pythagorean theorem to find the hypotenuse, and solve for the lengths of the sides of the triangle.

Step-by-step explanation:

To solve this problem, let's assign variables to the lengths of the two legs of the right triangle. Let the shorter leg be 'x' and the longer leg be '2x'.

The perimeter of a triangle is the sum of its three sides. So, we can write the equation: x + 2x + hypotenuse = 18.

Simplifying the equation, we get 3x + hypotenuse = 18. Rearranging, we have hypotenuse = 18 - 3x.

Now, we can use the Pythagorean theorem to find the length of the hypotenuse. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. So we have (x^2) + (2x^2) = (18 - 3x)^2.

Expanding and solving the equation, we find that x = 2. Substituting this value back into the equation, we can find the length of the two legs and the hypotenuse of the right triangle.

Answer 2

Let x represent the shortest leg of the triangle. If one leg twice as long as the other, then the length of the other leg is 2x.

Let y represent the hypotenuse which is the longest leg. The diagram is shown below

By applying pythagorean theorem,

y^2 = x^2 + (2x)^2

y^2 = x^2 + 4x^2 = 5x^2

Recall, perimeter of a triangle is the sum of the length of the sides of the triangle. If perimeter is 18, it means that

x + 2x + y = 18

3x + y = 18

y = 18 - 3x

Substituting y = 18 - 3x into y^2 = 5x^2, we have

(18 - 3x)^2 = 5x^2

324 - 54x - 54x + 9x^2 = 5x^2

9x^2 - 5x^2 - 54x - 54x + 324 = 0

4x^2 - 108x + 324 = 0

x^2 - 27x + 81 = 0

By applying the formula for quadratic equations, we have

`\begin{gathered} x\text{ = }\frac{-\text{ b }\pm\sqrt[]{b^2-4ac}}{2a} \\ x\text{ = }\frac{-\text{ - 27}\pm\sqrt[]{-27^2-4(1*81)}}{2*1} \\ x\text{ = }\frac{27\pm\sqrt[]{405}}{2} \\ x\text{ = }\frac{27\text{ + 20.12}}{2}\text{ or x = }\frac{27\text{ - 20.12}}{2} \\ x\text{ = 23.56 or x = 3.44} \end{gathered}`

The sides would be

3.44

2 x 3.44 = 6.88

18 - 3(3.44) = 7.68

The three sides are 3.44, 6.88 and 7.68