In a right-angled triangle ABC (angle C = 90), the height CD is drawn so that the length of the segment BD is 4 cm

| March 1, 2021 | education

In a right-angled triangle ABC (angle C = 90), the height CD is drawn so that the length of the segment BD is 4 cm longer than the length of the segment CD, AD = 9. Find the sides of triangle ABC.

Let the length of the segment CD = X cm, then, by condition, the length of the segment BD = (X + 4) cm.

Since CD is the height drawn to the hypotenuse from the vertex of the right angle, then:

CD ^ 2 = AD * BD.

X ^ 2 = 9 * (X + 4) = 9 * X + 36.

X ^ 2 – 9 * X – 36 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = (-9) ^ 2 – 4 * 1 * (-36) = 81 + 144 = 225.

X1 = (9 – √225) / (2/1) = (9 – 15) / 2 = -6 / 2 = -3. (Doesn’t fit because it’s less than 0).

X2 = (9 + √225) / (2/1) = (9 + 15) / 2 = 24/2 = 12 cm.

CD = 12 cm, then BD = 12 + 4 = 16 cm.

BA = BD + AD = 16 + 9 = 25 cm.

In a right-angled triangle CDA, CA ^ 2 = CD ^ 2 + AD ^ 2 = 144 + 81 = 225.

CA = 15 cm.

In a right-angled triangle ABC, BC ^ 2 = AB ^ 2 – CA ^ 2 = 625 – 225 = 400.

BC = 20 cm.

Answer: The sides of the ABC triangle are 15 cm, 20 cm, 25 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.