A right triangle has a hypotenuse of length 10√2 and an angle of 45 degrees, with a side opposite this angle with a length of 10. A second right triangle also has an angle of 45 degrees, with a side oppoite this angle with a length of 14. determine the length of the hypotenuse in the second triangle
To find the length of the hypotenuse in the second triangle, we can use the Pythagorean theorem. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In the first triangle, the length of the hypotenuse is given as 10√2 and the length of one side is 10. Let's call the length of the remaining side x. So, according to the Pythagorean theorem, we have:
(10√2)^2 = 10^2 + x^2
200 = 100 + x^2
x^2 = 100
Taking the square root of both sides, we get x = 10.
In the second triangle, the length of one side is given as 14. Let's call the length of the remaining side y. So, according to the Pythagorean theorem, we have:
hypotenuse^2 = 14^2 + y^2
hypotenuse^2 = 196 + y^2
Since the angle opposite to the hypotenuse is also 45 degrees in the second triangle, we can use the same values obtained from the first triangle: x = 10 and hypotenuse = 10√2. Substituting these values into the Pythagorean equation for the second triangle, we get:
(10√2)^2 = 196 + 10^2
200 = 196 + 100
Rearranging the equation, we find:
hypotenuse^2 = 296
Finally, taking the square root of both sides, we find:
hypotenuse = √296 = 2√74
Therefore, the length of the hypotenuse in the second triangle is 2√74.