Question

Решите, кто шарит. MN-хорда кола з центром у точці О, Р- середина MN. У цьому колі проведено радіуси ОN і OK, які проходять через точку Р. KNP=35°.Знайдіть кути трикутника PNO

Answer 1

KNP=KMP=35

Пусть KNM -вписанный = 1/2 дуги MK = 1/2 дуги KN тогда,

PON=KON - центральный KN=70 следовательно,

PNO=180-90-70=20

Ответ:OPN=90

Answer 2

MN-хорда кола з центром у точці О, Р- середина MN. У цьому колі проведено радіуси ОN і OK, які проходять через точку Р. KNP=35°. Знайдіть кути трикутника PNO.

Understanding the Problem

To solve this problem, we need to find the angles of triangle PNO. We are given that MN is a chord of a circle with center O, and R is the midpoint of MN. Radii ON and OK are drawn in this circle, passing through point R. We are also given that angle KNP is 35°.

Solution

To find the angles of triangle PNO, we can use the following steps:

1. Since ON and OK are radii of the circle, they are equal in length. Therefore, triangle ONK is an isosceles triangle. 2. In an isosceles triangle, the angles opposite the equal sides are also equal. Therefore, angle ONK is equal to angle OKN. 3. Since angle KNP is given as 35°, we can find angle ONK by subtracting 35° from 180° (the sum of angles in a triangle). 4. Once we find angle ONK, we can find angle OKN by using the fact that it is equal to angle ONK. 5. Finally, we can find angle PNO by subtracting the sum of angles ONK and OKN from 180°.

Let's calculate the angles step by step:

1. Angle ONK = 180° - angle KNP = 180° - 35° = 145°. 2. Angle OKN = angle ONK = 145°. 3. Angle PNO = 180° - (angle ONK + angle OKN) = 180° - (145° + 145°) = 180° - 290° = -110°.

Answer

The angles of triangle PNO are as follows: - Angle ONK = 145°. - Angle OKN = 145°. - Angle PNO = -110°.

Please note that the angle PNO is negative, which indicates that the triangle is not a valid triangle. It is not possible to have a negative angle in a triangle.