Question

точка k - середина боковой стороны cd трапеции abcd. прямая bk пересекается с ad в точке p. найдите длину отрезка ap, если основания трапеции равны 4 и 11

Answer 1

Треугольник ВСК = треугольнику ДКР, СК=КД, угол ДКР=уголСКВ как вертикальные, угол КДР=уголКСВ как внутренние разносторонние - по двум углам и стороне между ними, ДР=ВС=4, АР = АД+ДР=11+4=15

Answer 2

I can answer your question in detail. The point K is the midpoint of the lateral side CD of the trapezoid ABCD. The line BK intersects AD at point P. Find the length of the segment AP, if the bases of the trapezoid are equal to 4 and 11.

To solve this problem, we can use the properties of similar triangles and the midline theorem. Here is a diagram of the given situation:


Let x be the length of AP and y be the length of PD. Then, by the midline theorem, we have:

$$ BK = \frac{AB + CD}{2} = \frac{11 + 4}{2} = 7.5 $$

By the properties of similar triangles, we have:

$$ \frac{AP}{BK} = \frac{AD}{AB} \implies \frac{x}{7.5} = \frac{x + y}{11} $$

Solving for x, we get:

$$ x = \frac{82.5}{3.5} \approx 23.57 $$

Therefore, the length of the segment AP is approximately 23.57 units. You can find more details and explanations about this problem in the following sources: [1](https://geometria.my-dict.ru/q/3798916_v-trapecii-abcd-prodolzenia-bokovyh-storon/), [2](https://ege.sdamgia.ru/test?filter=all&category_id=276&ttest=true), [3](https://egemaximum.ru/putevoditel-po-zadacham-s4-16/), [4](https://online-otvet.ru/geometria/5b7497d8f0470558392b7163), [5](https://yandex.ru/tutor/subject/problem/?problem_id=T29479). I hope this helps.