Question

В остроугольном треугольнике АВС проведены высоты АК и ВР. а) Докажите, что углы АКР и АВР равны. б) Найдите длину отрезка РК, если известно, что АВ=5, ВС=6, СА=4. Заранее спасибо)

Answer 1

I will try to answer your question in detail. In an acute triangle ABC, the heights AK and BR are drawn.

a) To prove that the angles AKR and AVR are equal, we can use the following reasoning:

- Since AK and BR are heights, they are perpendicular to BC and AC respectively. Therefore, the angles AKC and BRC are right angles. - In the right triangles AKC and BRC, we have KC = RC, since they are the halves of the hypotenuse BC (by the median theorem). - We also have AC = AC, since it is a common side for both triangles. - By the SAS criterion, the triangles AKC and BRC are congruent. - Therefore, the corresponding angles are equal: AKR = AVR (by the CPCTC theorem).

b) To find the length of the segment RK, we can use the following formula:

- RK = (AB * BR) / AC - This formula can be derived from the similarity of the triangles ARK and ABC, and the fact that the ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides. - Substituting the given values, we get:

RK = (5 * BR) / 4

- To find BR, we can use the Pythagorean theorem in the right triangle BRC:

BR^2 + RC^2 = BC^2

- Substituting the given values, we get:

BR^2 + 3^2 = 6^2

- Simplifying, we get:

BR^2 = 27

- Taking the positive square root, we get:

BR = sqrt(27)

- Therefore, RK = (5 * sqrt(27)) / 4

- This is the exact value of RK. If you want to find an approximate value, you can use a calculator and get:

RK ≈ 4.13

I hope this helps you.