A right triangle has a hypotenuse of length 16 and an angle of 45°, with a side opposite this angle of length 82–√. A second right triangle also has an angle of 45° and a side opposite this angle with a length of 42–√. Determine the length of the hypotenuse in the second triangle.(1 point)
The hypotenuse of the second triangle has length 8.
The hypotenuse of the second triangle has length 8√2
The hypotenuse of the second triangle has length 4√2
The hypotenuse of the second triangle has length 4.
To solve this problem, we can use the property of right triangles that states that the side opposite the 45° angle is equal to the hypotenuse divided by the square root of 2.
In the first triangle, we are given that the hypotenuse is 16 and the side opposite the 45° angle is 82–√. Using the property mentioned above, we can set up the equation:
82–√ = 16/√2
To simplify this equation, we multiply both sides by √2:
(82–√) * √2 = 16
82√2 – √2 = 16
√2(82 – 1) = 16
√2(81) = 16
√2 = 16/81
This is our value for √2 in the first triangle.
Now, in the second triangle, we are given that the side opposite the 45° angle is 42–√. Again, using the property mentioned above, we can set up the equation:
42–√ = x/√2
To find x, the length of the hypotenuse, we multiply both sides by √2:
(42–√) * √2 = x
42√2 – √2 = x
√2(42 – 1) = x
√2(41) = x
√2 = x/41
This is our value for √2 in the second triangle.
Since √2 is the same in both triangles, we can set the two expressions equal to each other:
16/81 = x/41
Cross multiplying, we get:
41 * 16 = 81 * x
656 = 81x
Dividing both sides by 81, we find:
x = 656/81
We can simplify this fraction:
x = 8
Therefore, the length of the hypotenuse in the second triangle is 8.
The correct answer is:
The hypotenuse of the second triangle has a length of 8.