Question

кутова міра дуги МР дорівнює 68 градусів,а дуги РТ - 47 градусів. Знайдіть кут МРТ,якщо точки М,Р,Т лежать на одному колі так , що точка Р розташована між точками М і Т.

Answer 1

Градусная мера меньшей дуги MT равна $\tt\breve{MP}+\breve{PT}=68^\circ+47^\circ=115^\circ$

Тогда градусная мера большей дуги MT равна $\tt \breve{MT}=360^\circ-115^\circ=245^\circ$

Свойство вписанных углов:

• Вписанный угол равен половине дуги, на которую он опирается

То есть $\tt \angle MPT=\frac{1}{2} \breve{MT}=\frac{1}{2}\cdot245^\circ =122.5^\circ$

Ответ: 122,5°.

Answer 2

To find the angle MRT, we can use the fact that the sum of the angles in a triangle is 180 degrees. Since points M, R, and T lie on the same circle, we can consider the triangle MRT as an inscribed triangle. #### Inscribed Angle Theorem According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of its intercepted arc. In this case, the intercepted arc for angle MRT is arc MT. Given that the measure of arc MR is 68 degrees and the measure of arc RT is 47 degrees, we can find the measure of arc MT by subtracting the measures of arcs MR and RT from the total measure of the circle, which is 360 degrees. The measure of arc MT = 360 degrees - 68 degrees - 47 degrees = 245 degrees. Since angle MRT intercepts arc MT, the measure of angle MRT is half the measure of arc MT. #### Calculating the Angle MRT To find the measure of angle MRT, we divide the measure of arc MT by 2: Angle MRT = (245 degrees) / 2 = 122.5 degrees. Therefore, the measure of angle MRT is **122.5 degrees**. [[SOURCE 1]]