III gimnazjum

Matematyka 2001. Zeszyt ćwiczeń cz. 2

Matematyka

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Matematyka 2001

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Matematyka z plusem 3. Zbiór zadań 2001

Policzmy to razem 3

&nbsp;Pole trójkąta obliczamy jako połowę iloczynu podstawy i wysokości opuszczonej na tą podstawę.&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/a4b963e869a7e90de09201f58db9ab6e09999144b80f343ec3d654aa98c24fa7_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/a4b963e869a7e90de09201f58db9ab6e09999144b80f343ec3d654aa98c24fa7_big.svg" type="image/svg+xml"></object>

Proporcję rozwiązujemy mnożąc "na krzyż": <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/f39c6244776a39edbb5d92eac47b1dce7742d6bdb53f7c8a56a6d20b2987d4de_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/f39c6244776a39edbb5d92eac47b1dce7742d6bdb53f7c8a56a6d20b2987d4de_big.svg" type="image/svg+xml"></object>

Trójkąt równoboczny ma 3 boki jednakowej długości. Obliczmy, jaką długość ma jeden bok trójkąta:&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/993cb014e964cd5f647ea9f6547ffbf8c8fdd3aff7a12142483442f59944aa26_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/993cb014e964cd5f647ea9f6547ffbf8c8fdd3aff7a12142483442f59944aa26_big.svg" type="image/svg+xml"></object>

Podstawiamy kolejne równania i sprawdzamy.&nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/e53657178cb6855ac4b2029197a64b0ce6af712b69433254ae18be74d224f991_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/e53657178cb6855ac4b2029197a64b0ce6af712b69433254ae18be74d224f991_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/78f73d94ebf746925d3a0cf0d96c02c31eda7208b4e74622262f1de535cecde1_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/78f73d94ebf746925d3a0cf0d96c02c31eda7208b4e74622262f1de535cecde1_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/c5990e829d304a2af960ea23b3a578c59d2db2f20eaeb58bf805c179c7dc9288_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/c5990e829d304a2af960ea23b3a578c59d2db2f20eaeb58bf805c179c7dc9288_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/1484d6828a6c169163d5129b84da7c4b70cff21ad1d122df77011ab5eac5fa4d_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/1484d6828a6c169163d5129b84da7c4b70cff21ad1d122df77011ab5eac5fa4d_big.svg" type="image/svg+xml"></object> Otrzymaliśmy równość zawsze prawdziwą, więc układ jest nieoznaczony - ma nieskończenie wiele rozwiązań.&nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/be651d5a8e97690fbb1fafd9b89ea7bb5458b27b1d24a449c67ca762317ac1df_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/be651d5a8e97690fbb1fafd9b89ea7bb5458b27b1d24a449c67ca762317ac1df_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/a944b88321e4bf8b795bf60f563e75c4435c0cb685784914c3a2b144767d192a_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/a944b88321e4bf8b795bf60f563e75c4435c0cb685784914c3a2b144767d192a_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b06a02bf89ac48772705851e8c044cc599f22762f6085c95eb565994639c0433_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b06a02bf89ac48772705851e8c044cc599f22762f6085c95eb565994639c0433_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/536ff42686627a38b1b4a98fbd983b4a11d06e3af5ca5d1d8cd351b96a8817df_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/536ff42686627a38b1b4a98fbd983b4a11d06e3af5ca5d1d8cd351b96a8817df_big.svg" type="image/svg+xml"></object> Otrzymaliśmy sprzeczność, więc układ jest sprzeczny - nie ma rozwiązania.&nbsp; &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/acbbb708385721a6275edb28b2265071ab2c596d66c45e0f095ca22a55c6fb7d_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/acbbb708385721a6275edb28b2265071ab2c596d66c45e0f095ca22a55c6fb7d_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/3f73ba59aa2a2d0cd1763aa1a004ed730e6652086e23f972c38d02a272ce5256_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/3f73ba59aa2a2d0cd1763aa1a004ed730e6652086e23f972c38d02a272ce5256_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/10ad938725d7de14830163b6570f9cbc277752bbab023efdb5a4539aa7ac97d1_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/10ad938725d7de14830163b6570f9cbc277752bbab023efdb5a4539aa7ac97d1_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/738045d8d724cee5339c6c6e61bfac2a242fb97d9f317364c2cd21cb27bcd76c_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/738045d8d724cee5339c6c6e61bfac2a242fb97d9f317364c2cd21cb27bcd76c_big.svg" type="image/svg+xml"></object> Otrzymaliśmy sprzeczność, więc układ jest sprzeczny - nie ma rozwiązania.&nbsp; &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/f9e487799dfc7882b1a0db047999249489d02726b2c82055cb68a6bb0311f118_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/f9e487799dfc7882b1a0db047999249489d02726b2c82055cb68a6bb0311f118_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/da9bad7ab9b7b62ee75329c7b9f3f5fc801b6a695ce3dffb1b93e2fa2b8aab48_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/da9bad7ab9b7b62ee75329c7b9f3f5fc801b6a695ce3dffb1b93e2fa2b8aab48_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/64679cb2b704d04b969c405c0a4b2233a199a821d1a724cf63290dd7c6720b2e_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/64679cb2b704d04b969c405c0a4b2233a199a821d1a724cf63290dd7c6720b2e_big.svg" type="image/svg+xml"></object> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/19760621fd14d80d7acd11cc79ae48dc5bbb40731166e916ce075d043a6b9ab2_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/19760621fd14d80d7acd11cc79ae48dc5bbb40731166e916ce075d043a6b9ab2_big.svg" type="image/svg+xml"></object> Z powyższego równania można wyznaczyć y, potem wyliczoną wartość wstawić do dowolnego równania i wyliczyć x. Układ ma rozwiązanie, jest więc oznaczony.&nbsp; &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6a7b0a8d8a69a13d0c4f4d10baa07554326f3c9a2acfaa735c7b284971cc8ded_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6a7b0a8d8a69a13d0c4f4d10baa07554326f3c9a2acfaa735c7b284971cc8ded_big.svg" type="image/svg+xml"></object>

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