Question

what is the length of \\(\overline{ab}\\) in the accompanying figure?

the length of \\(\overline{ab}\\) is \\(square\\)
(type an exact answer, using radicals as needed.)

accompanying figure: a triangle with a perpendicular dashed line from the top vertex to base \\(ab\\), forming a right angle. the left side of the triangle is 10 ft, the right side is 17 ft, and the height (dashed line) is 8 ft.

Explanation:

Step1: Split into right triangles

The triangle is split by an 8ft height into two right triangles.

Step2: Calculate left segment of AB

For the left right triangle: hypotenuse=10ft, height=8ft. Use Pythagoras: $a^2 + 8^2 = 10^2$ → $a^2=100-64=36$ → $a=6$ ft.

Step3: Calculate right segment of AB

For the right right triangle: hypotenuse=17ft, height=8ft. Use Pythagoras: $b^2 + 8^2 =17^2$ → $b^2=289-64=225$ → $b=15$ ft.

Step4: Sum segments for AB length

AB = left segment + right segment = $6 + 15 = 21$ ft.

Answer:

21