{"paper":{"id":"site-2682","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","variant":null,"exam_pdf":null,"key_pdf":null,"question_count":20,"source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},"questions":[{"id":"site-2682/zad/2877","paper_id":"site-2682","number":"2877","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"W trójkącie równoramiennym ABC, w którym |AC|=|BC|=10 cm, wysokość poprowadzona z wierzchołka C jest równa 5 cm. Oblicz miary kątów tego trójkąta. Odpowiedź podaj w stopniach.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-2877.svg","solution_image":"img/site/2682/rozw-2877.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3214","paper_id":"site-2682","number":"3214","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Punkt D leży na boku BC trójkąta równoramiennego ABC, w którym |AC|=|BC|. Odcinek |AD| dzieli trójkąt ABC na dwa trójkąty równoramienne w taki sposób, że |AB|=|AD|=|CD| (patrz rysunek). Oblicz miary kątów trójkąta ABC.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3214.svg","solution_image":"img/site/2682/rozw-3214.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3215","paper_id":"site-2682","number":"3215","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Oblicz pole trójkąta równoramiennego ABC, w którym |AB|=24 i |AC|=|BC|=13.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3215.svg","solution_image":"img/site/2682/rozw-3215.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3216","paper_id":"site-2682","number":"3216","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Liczby 4, 10, c są długościami boków trójkąta równoramiennego. Oblicz c.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3216.svg","solution_image":"img/site/2682/rozw-3216.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3217","paper_id":"site-2682","number":"3217","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Liczby 6, 10, c są długościami boków trójkąta równoramiennego. Oblicz c.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3217.svg","solution_image":"img/site/2682/rozw-3217.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3218","paper_id":"site-2682","number":"3218","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Liczby 6, 10, c są długościami boków trójkąta prostokątnego. Oblicz c.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3218.svg","solution_image":"img/site/2682/rozw-3218.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3219","paper_id":"site-2682","number":"3219","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Obwód czworokąta wypukłego ABCD jest równy 50 cm. Obwód trójkąta ABD jest równy 46 cm, a obwód trójkąta BCD jest równy 36 cm. Oblicz długość przekątnej BD.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3219.svg","solution_image":"img/site/2682/rozw-3219.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3220","paper_id":"site-2682","number":"3220","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dany jest trójkąt prostokątny ABC, w którym |BC|=30, |AC|=40, |AB|=50. Punkt W jest środkiem okręgu wpisanego w ten trójkąt. Okrąg wpisany w trójkąt ABC jest styczny do boku AB w punkcie M. Oblicz długość odcinka CM.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3220.svg","solution_image":"img/site/2682/rozw-3220.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3221","paper_id":"site-2682","number":"3221","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Na zewnątrz trójkąta prostokątnego ABC, w którym |kąt ACB| = 90 stopni oraz |AC|=5, |BC|=12 zbudowano kwadrat ACDE (patrz rysunek). Punkt H leży na prostej AB i kąt |kąt EHA| = 90 stopni. Oblicz pole trójkąta HAE.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3221.svg","solution_image":"img/site/2682/rozw-3221.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/3304","paper_id":"site-2682","number":"3304","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Podstawy trapezu prostokątnego mają długość 6 i 10 oraz tangens kąta ostrego jest równy 3. Oblicz pole tego trapezu.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-3304.svg","solution_image":"img/site/2682/rozw-3304.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/5496","paper_id":"site-2682","number":"5496","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dany jest czworokąt ABCD , w którym |BC|=|CD|=|AD|=13 (zobacz rysunek). Przekątna BD tego czworokąta ma długość 10 i jest prostopadła do boku AD. Oblicz pole czworokąta ABCD.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-5496.svg","solution_image":"img/site/2682/rozw-5496.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/5791","paper_id":"site-2682","number":"5791","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Rozwiązanie zadania. Dany jest trójkąt równoramienny ABC, w którym podstawa AB ma długość 12, a każde z ramion AC i BC ma długość równą 10. Punkt D jest środkiem ramienia BC (zobacz rysunek). Oblicz sinus kąta α, jaki środkowa AD tworzy z ramieniem AC trójkąta ABC.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-5791.svg","solution_image":"img/site/2682/rozw-5791.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/6664","paper_id":"site-2682","number":"6664","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Odcinki AD i BC przecinają się w punkcie O. W trójkątach ABO i ODC zachodzą związki: |AO|=5, |BO|=3, |OC|=10, |∠OAB|=|∠OCD| (zobacz rysunek).","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-6664.svg","solution_image":"img/site/2682/rozw-6664.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/6731","paper_id":"site-2682","number":"6731","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"W trójkącie ABC dane są długości dwóch boków |AB|=12, |BC|=8 oraz miara kąta |∠ABC|=60°. Oblicz długość środkowej tego trójkąta, poprowadzonej z wierzchołka A.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-6731.svg","solution_image":"img/site/2682/rozw-6731.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/6732","paper_id":"site-2682","number":"6732","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Wierzchołki A i C trójkąta ABC leżą na okręgu o promieniu r. Środek S tego okręgu leży na boku AB tego trójkąta (zobacz rysunek poniżej). Długości boków AB i AC są równe odpowiednio |AB|=3r oraz |AC|=√3r. [rysunek] Oblicz miary wszystkich kątów trójkąta ABC.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-6732.svg","solution_image":"img/site/2682/rozw-6732.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/6735","paper_id":"site-2682","number":"6735","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dany jest trójkąt ABC. Na boku AB tego trójkąta wybrano punkt D, taki, że |AD|=1/4|AB|, a na boku BC wybrano taki punkt E, że |BE|=1/5|BC| (zobacz rysunek poniżej). Pole trójkąta ABC jest równe 20. [rysunek] Oblicz pole trójkąta DBE.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-6735.svg","solution_image":"img/site/2682/rozw-6735.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/6737","paper_id":"site-2682","number":"6737","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dany jest prostokąt ABCD, w którym |AD|=2. Kąt BDA ma miarę α, taką, że tgα=2. Przekątna BD i prosta przechodząca przez wierzchołek C prostopadła do BD przecinają się w punkcie E (zobacz rysunek). [rysunek] Oblicz długość odcinka |CE|.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-6737.svg","solution_image":"img/site/2682/rozw-6737.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/6740","paper_id":"site-2682","number":"6740","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Proste k i l przecinają się w punkcie A. Proste m, n i s są do siebie równoległe i przecinają obie proste k i l w punktach B,C,D,E,F,G (zobacz rysunek poniżej), w taki sposób, że: |BC|=30, |CD|=20, |GF|=21 [rysunek] Oblicz długość odcinka FE.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-6740.svg","solution_image":"img/site/2682/rozw-6740.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/6903","paper_id":"site-2682","number":"6903","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"Dany jest trapez równoramienny ABCD o podstawach AB i CD, gdzie |AB|>|CD|. Kąt ostry tego trapezu ma miarę 60°, a przekątna jest prostopadła do ramienia, którego długość jest równa 6. Oba ramiona tego trapezu przedłużono, otrzymując trapez DCFG podobny do trapezu ABCD (zobacz rysunek). [rysunek] Oblicz pole trapezu DCFG.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-6903.svg","solution_image":"img/site/2682/rozw-6903.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"},{"id":"site-2682/zad/8157","paper_id":"site-2682","number":"8157","points":null,"ptype":"open","subject":"matematyka","category":"zbiory","year":0,"month":null,"level":"podstawowa","text":"W trapezie równoramiennym ABCD przekątna BD ma długość 39 i tworzy z dłuższą podstawą AB taki kąt ostry α, że cosα = 12/13. Różnica długości podstaw AB i CD tego trapezu jest równa 40. Oblicz długość ramienia AD tego trapezu.","answer":null,"answer_text":null,"solution":null,"image":"img/site/2682/zad-8157.svg","solution_image":"img/site/2682/rozw-8157.svg","topics":null,"page_from":null,"source":"site","answer_source":null,"answer_text_source":null,"solution_source":null,"text_source":"site","source_label":"Matematyka · Zbiory zadań (podstawowa)","subject_label":"Matematyka","category_label":"Zbiory zadań"}]}