In a right-angled triangle ABC from the vertex C of the right angle, the height CD is drawn

| February 17, 2021 | education

In a right-angled triangle ABC from the vertex C of the right angle, the height CD is drawn, and in triangle ADC its bisector CE is drawn. Find the length of the segment BE, if it is known that AC = 6, BC = 8.

By the Pythagorean theorem, we determine the length of the hypotenuse AB.

AB ^ 2 = AC ^ 2 + BC ^ 2 = 36 + 64 = 100. AB = 10 cm.

Let the length of the segment AD = X cm, then ВD = (10 – X) cm.

Let us express the height of the СD in right-angled triangles ВСD and AСD.

СD ^ 2 = AC ^ 2 – AD ^ 2 = 36 – X2.

СD ^ 2 = BC ^ 2 – BD ^ 2 = 64 – (10 – X) 2.

Then: 36 – X2 = 64 – 100 + 20 * X – X2.

20 * X = 72.

X = AD = 72/20 = 3.6 cm.

Then BD = 10 – 3.6 = 6.4 cm.

Let’s determine the height of the CD. CD^2 = 36 – 3.62 = 23.04.

CD = 4.8 cm.

Let the length of the segment BE = X cm, then DE = (6.4 – X) cm.

In the BCD triangle, we use the property of the angle bisector.

BC / BE = CD / DE.

8 / X = 4.8 / (6.4 – X).

4.8 * X = 51.2 – 8 * X.

12.8 * X = 51.2.

X = BE = 51.2 / 12.8 = 4 cm.

Answer: The length of the segment BE is 4 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.