ABC is a triangle with AC=139 and BC=178. Points D and E are the midpoints of BC and ACrespectively. Given that AD and BE are perpendicular to each other, what is the length of AB?

101

process?

To find the length of AB, we can use the Pythagorean theorem. Let's break down the steps to solve this problem:

1. Draw a diagram: Sketch triangle ABC with AC = 139 and BC = 178. Label points D and E as the midpoints of BC and AC, respectively.

A
/ \
/ \
/ \
E-------D
/_________\
B C
2. Set up a right triangle: Since AD and BE are perpendicular, triangle ADE is a right triangle. Label AD = x and DE = y.

A
/ \
x / \ y
/ \
E-------D
/_________\
B C
3. Determine the lengths of AD and DE: Since D is the midpoint of BC, we know that BD = CD = 178/2 = 89. Similarly, since E is the midpoint of AC, AE = CE = 139/2 = 69. We can use these lengths to calculate the lengths of AD and DE using the Pythagorean theorem:

AD^2 = AE^2 + DE^2
x^2 = 69^2 + y^2

DE^2 + BD^2 = BE^2
y^2 + 89^2 = x^2

We have two equations and two variables. Let's solve this system of equations.
4. Solve the system of equations: Set the two equations equal to each other and substitute the values we know:

69^2 + y^2 = y^2 + 89^2

Simplify the equation by canceling out y^2 terms:
69^2 = 89^2

The equation simplifies to a false statement, which means there is no solution.
5. Interpret the result: Since the equation has no solution, it means that AD and BE cannot be perpendicular to each other. Therefore, there is not enough information to determine the length of AB in this case.

In summary, the length of AB cannot be determined with the given information because AD and BE cannot be perpendicular to each other.