Question

Точка K делит хорду AP на отрезки 12 и 14 см. Найдите радиус окружности, если расстояние от центра окружности до точки К равно 11 см.Заранее спасибо!

Answer 1

Чертеж во вложении.
В ∆РОК по теореме косинусов
$PO^2=PK^2+OK^2-2*PK*OK*cos\ \alpha$
В ∆АОК по теореме косинусов
$AO^2=AK^2+OK^2-2*AK*OK*cos(180^o-\alpha)$
Пусть ОА=ОР=r
Т.к. ∠ОКР и ∠ОКА - смежные, то $cos(180^o- \alpha)=-cos\ \alpha$
Приравняем правые части:
$AK^2+OK^2+2*AK*OK*cos \alpha=PK^2+OK^2-2*PK*OK*cos \alpha$
$AK^2+2*AK*OK*cos \alpha=PK^2-2*PK*OK*cos \alpha \\ 12^2+2*12*11*cos \alpha=14^2-2*14*11*cos \alpha\\ 2*11*(14+12)*cos \alpha=14^2-12^2\\ 22*26*cos \alpha=(14-12)(14+12)\\ 22*26*cos \alpha=2*26\\ cos\ \alpha= \frac{1}{11}\\ r^2=11^2+14^2-2*11*14*\frac{1}{11}=289\\ r=\sqrt{289}=17$
Ответ: 17.

Answer 2

Problem Analysis

We are given that point K divides chord AP into segments of 12 cm and 14 cm. The distance from the center of the circle to point K is 11 cm. We need to find the radius of the circle.

Solution

Let's assume the radius of the circle is r.

From the given information, we can form the equation: r^2 = 12 cm * 14 cm

Simplifying the equation: r^2 = 168 cm^2

Taking the square root of both sides: r = √(168) cm

Calculating the square root of 168: r ≈ 12.96 cm

Therefore, the radius of the circle is approximately 12.96 cm.

Answer

The radius of the circle is approximately 12.96 cm.