Question

Две окружности радиусов R=9, r=7 касаются внешним образом в точке A.. через точку B, взятую на большей окружности, проведена прямая, касающаяся меньшей окружности в точке С. Найдите длину отрезка ВС, если длина хорды AB равна 12. Решение прошу сделать с рисунком

Answer 1

Сделаем рисунок.
Пусть центр большей окружности будет М, меньшей - Н
Заметим сразу, что точка В, взятая на большей окружности, не является точкой касания прямой ВС и этой окружности, т.е. угол МВС - не прямой. .
ВС можем найти из прямоугольного треугольника ВСН, где  СН=R=7. 
ВН можно найти по т. косинусов из треугольника ВМН, в котором известны две стороны, а косинус угла ВМН можно найти. 
Рассмотрим треугольник АМВ. 
По т.косинусов АВ²=ВМ²+АМ² -2*ВМ*АМ*cos АМВ
144=81+81-2*81*cos АМВ
-18=- 162*cos АМВ
cos АМВ=1/9
В треугольнике ВМН 
ВН²=ВМ²+МН²-2*ВМ*ВН*cos НМВ
cos НМВ=cos АМВ=1/9
МН=9+7=16
ВН²=9²+16²-2*9*16*1/9
ВН²=305
Из треугольника ВСН
ВС²=ВН²-СН²
ВС=√(305-49)
BC=√256=16

Answer 2

Problem Analysis

We are given two circles with radii R = 9 and r = 7, which are externally tangent at point A. A line is drawn through point B, taken on the larger circle, which is tangent to the smaller circle at point C. We need to find the length of segment BC, given that the length of chord AB is 12.

Solution

To solve this problem, we can use the properties of tangents and chords in circles. Let's break down the solution into steps:

1. Draw a diagram to visualize the problem. We have two circles, one larger with radius R = 9 and one smaller with radius r = 7. The circles are externally tangent at point A. Point B is taken on the larger circle, and a line is drawn through B, tangent to the smaller circle at point C.

2. Let's label the points on the diagram. We have point A as the point of tangency between the two circles, point B on the larger circle, and point C on the smaller circle.

3. We are given that the length of chord AB is 12. Let's label the length of AB as 12.

4. Since the line through B is tangent to the smaller circle at point C, we know that BC is perpendicular to AB. This means that triangle ABC is a right triangle.

5. We can use the Pythagorean theorem to find the length of BC. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

6. In triangle ABC, BC is the hypotenuse, and AB and AC are the other two sides. We know that AB = 12, and we need to find the length of BC.

7. Applying the Pythagorean theorem, we have BC^2 = AB^2 - AC^2.

8. To find AC, we can use the fact that AC is the radius of the smaller circle, which is r = 7.

9. Substituting the values, we have BC^2 = 12^2 - 7^2.

10. Simplifying, we have BC^2 = 144 - 49.

11. Therefore, BC^2 = 95.

12. Taking the square root of both sides, we have BC = √95.

13. The length of segment BC is approximately 9.747 units.

Conclusion

The length of segment BC, given that the length of chord AB is 12, is approximately 9.747 units.