В треугольнике  S P R SPR проведена биссектриса  S Q SQ. Найди  R Q RQ, если  S R = 16 , 5 SR=16,5,  S P = 10 SP=10,  Q P = 8 QP=8.

Для начала найдем длину отрезка PR с помощью теоремы косинусов:

PR^2 = SR^2 + SP^2 - 2 * SR * SP * cos(S)

PR^2 = 16.5^2 + 10^2 - 2 * 16.5 * 10 * cos(S)

PR^2 = 272.25 + 100 - 330 * cos(S)

Также, по теореме угловой биссектрисы:

SQ/SP = SR/PR

SQ = SR * SP / PR

SQ = 16.5 * 10 / sqrt(272.25 + 100 - 330 * cos(S))

Теперь найдем длину отрезка RQ с помощью теоремы косинусов в треугольнике SPR:

RQ^2 = SR^2 + SQ^2 - 2 * SR * SQ * cos(S)

RQ^2 = 16.5^2 + SQ^2 - 2 * 16.5 * SQ * cos(S)

RQ^2 = 272.25 + SQ^2 - 33 * SQ * cos(S)

RQ = sqrt(272.25 + SQ^2 - 33 * SQ * cos(S))

Подставляем найденное значение SQ и получаем:

RQ = sqrt(272.25 + (16.5 * 10 / sqrt(272.25 + 100 - 330 * cos(S)))^2 - 33 * (16.5 * 10 / sqrt(272.25 + 100 - 330 * cos(S))) * cos(S))

RQ = sqrt(272.25 + (165 / sqrt(272.25 + 100 - 330 * cos(S)))^2 - 33 * (165 / sqrt(272.25 + 100 - 330 * cos(S))) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 33 * 165 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 - 330 * cos(S)) * cos(S))

RQ = sqrt(272.25 + 272.25 / (272.25 + 100 - 330 * cos(S)) - 5445 / (272.25 + 100 -