Question

Помогите пожалуйста решить В треугольнике KLM стороны KM =17, LM =13, KL =18. Окружность, проходящая через точки K и L, пересекает стороны треугольника MK и ML соответственно в точках P и Q . Отрезок PQ лежит внутри треугольника KLM и касается окружности, вписанной в треугольник KLM. Найдите длину отрезка PQ. Варианты ответа: 2,5 3,25 10 4,5

Answer 1

Треугольники PMQ и LMK подобны т.к. KPQL - вписанный). Значит
18/PQ=17/(13-QL)=13/(17-KP).
KP+QL=PQ+18, т.к. KPQL - описанный. Решаем эту систему, получаем PQ=9/2.

Answer 2

Problem Analysis

We are given a triangle KLM with side lengths KM = 17, LM = 13, and KL = 18. An inscribed circle in triangle KLM intersects the sides MK and ML at points P and Q, respectively. We need to find the length of segment PQ.

Solution

To find the length of segment PQ, we can use the property that the lengths of tangents drawn from an external point to a circle are equal. In this case, segment PQ is tangent to the inscribed circle.

Let's denote the length of segment PQ as x.

Since segment PQ is tangent to the inscribed circle, we can draw radii from the center of the circle to points P and Q. Let's denote the center of the circle as O.

Now, we have two right triangles: triangle KOP and triangle LQO. The lengths of the legs of these triangles are equal to the radii of the inscribed circle, which we can denote as r.

Using the Pythagorean theorem, we can write the following equations:

For triangle KOP: KP^2 + OP^2 = KO^2

For triangle LQO: LQ^2 + OQ^2 = LO^2

Since KP = LQ (tangents drawn from an external point to a circle are equal), we can rewrite the equations as:

KP^2 + OP^2 = KO^2 KP^2 + OQ^2 = LO^2

Substituting the values of the lengths of the sides of triangle KLM, we can write:

KP^2 + r^2 = (KM - r)^2 KP^2 + r^2 = (LM - r)^2

Simplifying these equations, we get:

KP^2 + r^2 = KM^2 - 2KM*r + r^2 KP^2 + r^2 = LM^2 - 2LM*r + r^2

Canceling out the common terms, we have:

KP^2 = KM^2 - 2KM*r KP^2 = LM^2 - 2LM*r

Since KP = LQ, we can equate the two equations:

KM^2 - 2KM*r = LM^2 - 2LM*r

Substituting the values of KM, LM, and KL, we can solve for r:

17^2 - 2*17*r = 13^2 - 2*13*r

289 - 34r = 169 - 26r

8r = 120

r = 15

Now that we have the value of r, we can find the length of segment PQ:

PQ = KP + KQ = KP + LQ = 2KP

Substituting the value of KP from the equation KP^2 = KM^2 - 2KM*r, we get:

KP = sqrt(KM^2 - 2KM*r)

Substituting the values of KM and r, we have:

KP = sqrt(17^2 - 2*17*15)

KP = sqrt(289 - 510)

KP = sqrt(-221)

Since the value under the square root is negative, it means that the triangle KLM is not constructible with the given side lengths. Therefore, we cannot find the length of segment PQ.

Answer: The length of segment PQ cannot be determined with the given side lengths of triangle KLM.