The height drawn to the hypotenuse of a right-angled triangle is 5 cm, and the length of one
The height drawn to the hypotenuse of a right-angled triangle is 5 cm, and the length of one of the legs is 13 cm. Calculate the length of the hypotenuse of the triangle.
The height of the CH forms two right-angled triangles АСН and ВСН.
From the right-angled triangle ACH, according to the Pythagorean theorem, we determine the length of the leg AH.
AH ^ 2 = AC ^ 2 – CH ^ 2 = 169 – 25 = 144.
AH = 12 cm.
Let us prove the similarity of triangles ACH and BCH.
Let the value of the angle HAC of the triangle ABC be equal to X0, then the angle ACH = (90 – X) 0.
Angle АСВ = 90, then angle ВСН = (90 – (90 – X) = X0.
The acute angles of the right-angled triangles АСН and ВСН are equal, then the triangles are similar in acute angle.
Then in similar triangles АН / СН = СН / ВН.
CH ^ 2 = AH * BH.
BH = CH ^ 2 / AH = 25/12 = 2 (1/12) cm.
Then AB = AH + BH = 12 + 2 (1/12) = 169/12 = 14 (1/12) cm.
Answer: The length of the hypotenuse AB is 14 (1/12) cm.