# The height CD is drawn in a right-angled triangle with a right angle c. It is known that BD = 16cm

The height CD is drawn in a right-angled triangle with a right angle c. It is known that BD = 16cm, CD = 12cm. Find AB, CB, AC, AD.

In a right-angled triangle BCD, according to the Pythagorean theorem, we determine the length of the hypotenuse BC.

ВС ^ 2 = ВD ^ 2 + СD ^ 2 = 256 + 144 = 400.

BC = 20 cm.

Let us prove that triangles ВСD and АСD are similar.

Let the angle CAD = X0, then the angle ACD = (90 – X) 0.

In a triangle ВСD, the angle ВСD = (90 – АСD) = (90 – (90 – X) = X0.

Then triangles ВСD and АСD are similar in acute angle.

Then, in similar triangles:

CD / BD = AD / CD.

CD ^ 2 = AD * BD.

AD = CD ^ 2 / BD = 144/16 = 9 cm.

Then AB = AD + BD = 9 + 16 = 25 cm.

In a right-angled triangle CAD, according to the Pythagorean theorem, AC ^ 2 = CD ^ 2 + AD ^ 2 = 144 + 81 = 225.

AC = 15 cm.

Answer: AB = 25 cm, CB = 20 cm, AC = 15 cm, AD = 9 cm.

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