Answer: The hypotenuse are both 13 cm
Explanation:
The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
So, in this case:
c^2 = a^2 + b^2
c^2 = 5^2 + 12^2
c^2 = 25 + 144
c^2 = 169
c = √169
c = 13
So the length of the hypotenuse is 13 cm.
Next, we can use the Pythagorean theorem to find the length of the segments into which the bisector of the right angle divides the hypotenuse. Let's call the length of the segments x. Then, the segments form two right triangles with legs of length x and (13-x)/2.
Using the Pythagorean theorem on each triangle, we have:
x^2 + ((13-x)/2)^2 = 13^2
x^2 + (13-x)^2/4 = 169
x^2 + (13^2 - 2x(13) + x^2)/4 = 169
x^2 + (13^2 - 2x(13))/4 = 169
x^2 + (169 - 26x)/4 = 169
x^2 + 169 - 26x = 676
x^2 - 26x + 507 = 0
Now, we can use the quadratic formula to find x:
x = (-b ± √(b^2 - 4ac)) / 2a
a = 1, b = -26, c = 507
x = (26 ± √(26^2 - 4 * 1 * 507)) / 2 * 1
x = (26 ± √(676)) / 2
x = (26 ± 26) / 2
x = 26 / 2
x = 13
So, the lengths of the segments into which the bisector of the right angle divides the hypotenuse are both 13 cm, to the nearest tenth.