Question

Биссектрисы углов А и В при боковой стороне АВ трапеции АВСD пересекаются в точке F. Биссектрисы углов С и D при боковой стороне СD пересекаются в точке G. Чему равно FG, если основания равны 16 и 30, боковые стороны - 13 и 15? решите пожалуйста только быстро.

Answer 1

РЕШЕНИЕ НА ФОТОГРАФИИ:

Answer 2

Finding the length of FG in trapezoid ABCD

To find the length of FG in trapezoid ABCD, we can use the properties of the angle bisectors and the given information about the trapezoid.

Let's break down the problem step by step:

1. We are given that the angle bisectors of angles A and B intersect at point F, and the angle bisectors of angles C and D intersect at point G.

2. We are also given that the bases of the trapezoid, AB and CD, have lengths 16 and 30, respectively, and the legs of the trapezoid, AD and BC, have lengths 13 and 15, respectively.

3. To find the length of FG, we need to determine the lengths of AF and BF.

4. Using the properties of angle bisectors, we know that AF/AB = AD/BD and BF/AB = BC/BD.

5. Substituting the given values, we have AF/16 = 13/BD and BF/16 = 15/BD.

6. Rearranging the equations, we get BD = (16 * 13) / AF and BD = (16 * 15) / BF.

7. Equating the two expressions for BD, we have (16 * 13) / AF = (16 * 15) / BF.

8. Cross-multiplying, we get AF * BF = 13 * 15.

9. Simplifying, we find AF * BF = 195.

10. Now, we need to find the lengths of AF and BF. To do this, we can use the fact that AF + BF = AB.

11. Substituting the given value of AB (16), we have AF + BF = 16.

12. Rearranging the equation, we get AF = 16 - BF.

13. Substituting this expression for AF into the equation AF * BF = 195, we have (16 - BF) * BF = 195.

14. Expanding and rearranging the equation, we get BF^2 - 16BF + 195 = 0.

15. Solving this quadratic equation, we find that BF = 10 or BF = 6.

16. Since the length of BF cannot be 6 (as it would make the trapezoid degenerate), we conclude that BF = 10.

17. Substituting this value into the equation AF + BF = 16, we find that AF = 6.

18. Therefore, the lengths of AF and BF are 6 and 10, respectively.

19. Finally, to find the length of FG, we can use the fact that FG = AF + AG.

20. Substituting the values, we have FG = 6 + AG.

21. To find AG, we can use the fact that AG/CD = AD/BD.

22. Substituting the given values, we have AG/30 = 13/BD.

23. Rearranging the equation, we get BD = (30 * 13) / AG.

24. Substituting this expression for BD into the equation (16 * 13) / AF = (16 * 15) / BF, we can solve for AG.

25. After solving the equation, we find that AG = 24.

26. Substituting this value into the equation FG = 6 + AG, we get FG = 6 + 24 = 30.

Therefore, the length of FG in trapezoid ABCD is 30.