Find the height of a right-angled triangle drawn from the top of the right angle and dividing its hypotenuse

Find the height of a right-angled triangle drawn from the top of the right angle and dividing its hypotenuse into 3cm and 3cm segments.

Given: triangle ABC, angle C – straight line, CH – height. AH = HB = 3cm.

Find: CH.

Decision:

1) Since AH = HB = 3 cm (by condition), CH is the median. But CH is also a height, therefore, triangle ABC is isosceles. AC = CB.

2) By the Pythagorean theorem, AC ^ 2 + CB ^ 2 = AB ^ 2,

AB = AH + HB = 6 cm,

AC = CB.

Hence, AC ^ 2 + AC ^ 2 = 36.

2 * AC ^ 2 = 36,

AC ^ 2 = 18.

3) On the other hand, triangle ACH is rectangular since CH is height. By the Pythagorean theorem, CH ^ 2 + AH ^ 2 = AC ^ 2,

CH ^ 2 + AH ^ 2 = 18,

AH = 3 cm,

CH ^ 2 = 18 – 9,

CH ^ 2 = 9,

CH = 3 cm.

Answer: CH = 3 cm.