Дана трапеция ABCD,диагонлиAC,BD.AO=14,OC=6,BO=x,OD=y,AC=20,BD=32.Найти BO,OD

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Ответ:

Gjjsg

01.02.2020 22:22

Обозначим длину стороны AB за x (x ≥ 0). Вспомним формулу нахождения описанной около треугольника окружности через произведение сторон и площадь
$R = \frac{AB \cdot BC \cdot AC}{4S_{\Delta ABC}}$

$\frac8{\sqrt{15}} = \frac{3 \cdot 4 \cdot x}{4S}$
$\frac8{\sqrt{15}} = \frac{3 \cdot x}{S}$
$8S=3x\sqrt{15}$

Найдем площадь треугольника по формуле Герона
$S=\sqrt{p(p-AB)(p-AC)(p-BC)}$ , где $p=\frac{AB+AC+BC}2$

$p=\frac{3+4+x}2=\frac{7+x}2$

$S=\sqrt{\frac{7+x}2(\frac{7+x}2-3)(\frac{7+x}2-4)(\frac{7+x}2-x)}=$
$=\sqrt{\frac{7+x}2\cdot\frac{1+x}2\cdot\frac{x-1}2\cdot\frac{7-x}2}=\sqrt{(\frac72+\frac x2)(\frac72-\frac x2)(\frac x2+\frac12)(\frac x2-\frac12)}=$
$\sqrt{(\frac{49}4-\frac{x^2}4)(\frac{x^2}4-\frac14)}=\frac14\sqrt{(49-x^2)(x^2-1)}$

Подставим получившееся значение в первое уравнение
$8\cdot\frac14\sqrt{(49-x^2)(x^2-1)}=3x\sqrt{15}$
$2\sqrt{(49-x^2)(x^2-1)}=3x\sqrt{15}$
$(2\sqrt{(49-x^2)(x^2-1)})^2=(3x\sqrt{15})^2$
$4(49-x^2)(x^2-1)=9x\cdot15$
196x^2-196-4x^4+4x^2=135x

Замена $x^2=t,\ t \geq 0$

4t^2-65t+196=0

$D=65^2-4\cdot4\cdot196=4225-3136=1089=33^2$
$t_1=\frac{65+33}{2\cdot4}=12,25$
$t_2=\frac{65-33}{2\cdot4}=4$

Вернемся к замене
$1)\ x^2=12,25$
$x=\pm3,5$
$2)\ x^2=4$
$x=\pm2$
$x \geq 0 \Rightarrow x \in \{3,5;\ 2\}$

Найдем больший угол треугольника по теореме косинусов
1) Стороны: 3; 4; 3,5
$A{C^2} = B{C^2} + A{B^2} - 2 \cdot BC \cdot AB \cdot \cos \angle B$
$4^2 = 3,5^2 + 3^2 - 2 \cdot 3,5 \cdot 3 \cdot \cos \angle B
$
$16 = 12,25 + 9 - 21\cos \angle B
$
$21\cos \angle B=5,25
$
$\cos \angle B=0,25$
Значит ∠B < 90° ⇒ ΔABC - остроугольный.

2) Стороны: 3; 4; 2
$A{C^2} = B{C^2} + A{B^2} - 2 \cdot BC \cdot AB \cdot \cos \angle B$
$4^2 = 2^2 + 3^2 - 2 \cdot 2 \cdot 3 \cdot \cos \angle B$
$16 = 4 + 9 - 12\cos \angle B$
$12\cos \angle B =-3
$
$\cos \angle B =-0,25
$
Значит ∠B > 90° ⇒ ΔABC - тупоугольный.

По условию треугольник тупоугольный, значит AB = 2, а P = 3 + 4 + 2 = 9

ответ: 9

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Ответ:

Max2934

15.02.2023 13:58

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2.we didn't go to Moscow last summer.

the children didn't have dinner at 2o'clock yesterday.

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3.where the children play in the garden yesterday?

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4.1 We didn't have lunch at 1o'clock yesterday.Did we have lunch at 1o'clock yesterday? where we have lunch at 1 o'clock yesterday? Who have lunch at 1 o'clock yesterday?

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5/2 where you live last month?

5/3 Did they wake up at 7 am ?

5/4 We didn't go to school yesterday

5/5Who watch the TV on Sunday ?

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