Answer:
The perimeter of the triangle is 30.5.
Explanation:
To find the perimeter of the triangle, we need to find the length of the other leg.
We can use the equation of the line that defines the hypotenuse, y = 3x + 2, to find the coordinate of the endpoint of the shorter leg. The endpoint has x-coordinate equal to 3 (since the length of the leg is 3), and the y-coordinate can be found by substituting x = 3 into the equation:
y = 3x + 2
y = 3 * 3 + 2
y = 11
So the endpoint of the shorter leg is (3, 11).
Next, we can use the Pythagorean theorem to find the length of the other leg:
a^2 + b^2 = c^2
where a is the length of the shorter leg (which we know is 3), b is the length of the other leg, and c is the length of the hypotenuse (which is also on the line y = 3x + 2).
We can use the coordinates of the endpoint of the shorter leg to find the length of the hypotenuse:
c = √((3 - 0)^2 + (11 - 2)^2)
c = √(3^2 + 9^2)
c = √(9 + 81)
c = √90
So the length of the hypotenuse is √90.
Using the Pythagorean theorem, we can now find the length of the other leg:
3^2 + b^2 = (√90)^2
9 + b^2 = 90
b^2 = 81
b = 9
So the length of the other leg is 9.
Finally, the perimeter of the triangle is 3 + 9 + √90, which is approximately 12 + 9 + 9.5 = 30.5.