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 Suppose that ABC is a right triangle with right angle at B. If AC = 25 and the altitude BD = 20, what is AD?

 Feb 27, 2020
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Draw right triangle(ABD) with angle(B) the right angle. Therefore, AC will be the hypotenuse.  AC = 25.

Draw altitude BD. BD = 20

Since B is on AC, let AD = x and DC = 25 - x.

 

Triangle(ADB) is a right triangle with angle(D) the right angle.

Using the Pythagorean Theorem:  AD2 + DB2 = AB2   --->   AB2 = x2 + 202.

 

Triangle(BDC) is a right triangle with angle(D) the right angle.

Using the Pythagorean Theorem:  CD2 + DB2 = CB2   --->   CB2 = (25 - x)2 + 202.

 

Using the Pythagorean Theorem on right triangle(ABC):  AB2 + CB2 = AC2,

Substituting:  [x2 + 202] + [(25 - x)2 + 202] = 252

Simplifying:  x2 + 400 + 625 -50x + x2 + 400 = 625

--->                                      2x2 - 50x +1425 = 625

--->                                       2x2 - 50x + 800 = 0

--->                                          x2 - 15x + 400 = 0

 

This equation has no real roots, so the original problem is impossible to occur.

 Feb 27, 2020

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