Final answer:
After setting BD equal to AD and DC, and solving the equations, we find that x = 10 and y = 2.
Consequently, BD, AD, and DC measure 13 units each, and the hypotenuse AC measures 26 units.
Step-by-step explanation:
We are given a right triangle ABC with BD as the median to the hypotenuse AC.
The lengths of segments are given as BD = x + 3, AD = 6y + 1, and DC = x + y + 1.
By the properties of a median to the hypotenuse in a right triangle, we know that BD = AD = DC.
Thus, setting BD equal to AD and BD equal to DC provides us with two equations:
- x + 3 = 6y + 1
- x + 3 = x + y + 1
Solving the first equation for y gives us y = (x + 2)/6.
Solving the second equation for y gives us y = 2.
Substituting y = 2 into the first equation, we get x = 6(2) - 2 = 10.
Now, calculating the lengths:
BD = x + 3
= 10 + 3
= 13
AD = 6y + 1
= 6(2) + 1
= 13
DC = x + y + 1
= 10 + 2 + 1
= 13
Since BD is the median to the hypotenuse, AC is twice the length of BD:
AC = 2 * BD
= 2 * 13
= 26