Suppose that the endpoints of one leg of a 45°-45°-90° triangle are (0, 3) and (4, –1). What is the length of the hypotenuse?
The distance between the two points is √(16+16) = √32
So, the hypotenuse is √32*√2 = √64 = 8
Find the distance EA for the points E(2, 3) and A(4, 2).
To find the length of the hypotenuse of a 45°-45°-90° triangle, we can use the properties of right triangles.
Let's start by finding the length of the leg of the triangle using the given endpoints. We can use the distance formula to find the length of a line segment between two points in a coordinate plane.
Distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the given endpoints (0, 3) and (4, -1), we can substitute the values into the distance formula:
d = sqrt((4 - 0)^2 + (-1 - 3)^2)
= sqrt(4^2 + (-4)^2)
= sqrt(16 + 16)
= sqrt(32)
= 4sqrt(2)
So, the length of one leg of the 45°-45°-90° triangle is 4√2.
In a 45°-45°-90° triangle, the legs are congruent. Therefore, the other leg of the triangle will also have a length of 4√2.
To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Pythagorean theorem:
c^2 = a^2 + b^2
Since the triangle is a 45°-45°-90° triangle, the two legs are congruent, so we can substitute the length of one leg (4√2) into the formula:
c^2 = (4√2)^2 + (4√2)^2
c^2 = 16(2) + 16(2)
c^2 = 32 + 32
c^2 = 64
Taking the square root of both sides, we find:
c = sqrt(64)
c = 8
Therefore, the length of the hypotenuse of the 45°-45°-90° triangle is 8.