In a right-angled triangle with leg 9, the length of the height drawn to the hypotenuse is 18.

In a right-angled triangle with leg 9, the length of the height drawn to the hypotenuse is 18. Find the length of the hypotenuse.

Given:
Right-angled triangle ABC,
AC = 9√5,
CK – height,
CK = 18.
Find the length of the hypotenuse AB -?
Solution:
1) Consider the triangle ACK according to the Pythagorean theorem:
AK² = (9√5) ² – 18²;
AK² = 405 – 324;
AK² = 81;
AK = 9.
2) Consider the ABC triangle:
CK² = AK · KB;
18² = 9 · BK;
324 = 9 BK;
BK = 324: 9;
BK = 36
3) AB = AK + BK = 9 + 36 = 45.
Answer: 45.



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