Question

Alice wants to put a fence around a portion of her yard in the shape of a right triangle. She knows that the larger leg of the triangle is

2
feet longer than its smaller leg. The hypotenuse is
4
feet longer than the smaller leg.

How many feet long is the hypotenuse?

Answer 1

By using the Pythagorean theorem and setting the length of the smaller leg as x, we find that x equals 6, leading to a hypotenuse of x + 4, which calculates to 10 feet.

Step-by-step explanation:

Let's call the length of the smaller leg of the triangle x feet. According to the problem, the larger leg is x + 2 feet and the hypotenuse is x + 4 feet. By applying the Pythagorean theorem, which is a² + b² = c², where a and b are the legs and c is the hypotenuse of a right triangle, we can find the length of the hypotenuse.

Let's substitute the known values into the equation:

x² + (x + 2)² = (x + 4)²

Now, we solve for x:

• x² + x² + 4x + 4 = x² + 8x + 16
• 2x² + 4x + 4 = x² + 8x + 16
• x² - 4x - 12 = 0

Factoring the quadratic equation:

• (x - 6)(x + 2) = 0

So, x can be either 6 or -2. Since the length of a side cannot be negative, the smaller leg is 6 feet. Applying this to the hypotenuse:

Hypotenuse = Smaller leg + 4 = 6 feet + 4 feet = 10 feet

Therefore, the hypotenuse is 10 feet long.

Answer 2

Step-by-step explanation:

`h^2=x^2+y^2\\ \\ h=y+4,\ x=y+2\\ \\ (y+4)^2=(y+2)^2+y^2\\ \\ y^2+8y+16=y^2+4y+4+y^2\\ \\ y^2+8y+16=2y^2+4y+4\\ \\ y^2-4y-12=0\\ \\ (y-6)(y+2)=0\\ \\ y>0\\ \\ y=6ft\\ \\ h=6+4=10ft`