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MATeMAtyka 3. Zakres podstawowy i rozszerzony. Po gimnazjum

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<img class="img-responsive img-thumbnail dropzone-img" src="https://odrabiamy.pl//uploads/exercise_photo/image/105064/16_20159.jpg"> Zauważmy, że&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2359b4aae8121c3ff6377b1c8a7f828aa3d9efbd7f3e3e96bdb297450995102e_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2359b4aae8121c3ff6377b1c8a7f828aa3d9efbd7f3e3e96bdb297450995102e_big.svg" type="image/svg+xml"></object> &nbsp; Czyli trójkąt ADB jest to trójkąt równoboczny. Oznaczmy&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8d417da3d84edee04cd249ecad1308c544d180e87137f9d76e3181968feaa140_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8d417da3d84edee04cd249ecad1308c544d180e87137f9d76e3181968feaa140_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/e78d5c9e207f862de27c6732f670a0ffbaf039b406634bc30f21cd49d4ea5517_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/e78d5c9e207f862de27c6732f670a0ffbaf039b406634bc30f21cd49d4ea5517_big.svg" type="image/svg+xml"></object> &nbsp; Z podobieństwa trójkątów CDE i ABE otrzymujemy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/be7ac580f78de88fa8746c21112e3f1b5d191144c3082fdc0da67e9034d6f159_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/be7ac580f78de88fa8746c21112e3f1b5d191144c3082fdc0da67e9034d6f159_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/69693e0315c4a152d96ffd47ddd5af7179389f1bd54354e0121fd8c9aeb6bf85_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/69693e0315c4a152d96ffd47ddd5af7179389f1bd54354e0121fd8c9aeb6bf85_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/377b3a573a53f613cc062f07d9e2cd6a935b6c4652aaf5bf5085b801e1f70375_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/377b3a573a53f613cc062f07d9e2cd6a935b6c4652aaf5bf5085b801e1f70375_big.svg" type="image/svg+xml"></object>

Zauważmy, że w podstawie mamy trójkąt równoboczny. Promień okręgu wpisanego w trójkąt równoboczny wyraża się wzorem&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/4d431923abc9260d82704e2ca3bfc7fe0a11422cd80fc258beccaaa804733559_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/4d431923abc9260d82704e2ca3bfc7fe0a11422cd80fc258beccaaa804733559_big.svg" type="image/svg+xml"></object> &nbsp; Wobec tego&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/db8d0563f766cfb644cc349f2bb5da2854100c36104176a56e7af8aff42e8b52_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/db8d0563f766cfb644cc349f2bb5da2854100c36104176a56e7af8aff42e8b52_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8e34f657aa6ea4a72e23193aaa12183057b87556e410f36ad35a686522837cc2_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8e34f657aa6ea4a72e23193aaa12183057b87556e410f36ad35a686522837cc2_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2926001ea6d3f4c592ad985282ca0621a4808cd2be2ffd506c4bb007527a44a1_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2926001ea6d3f4c592ad985282ca0621a4808cd2be2ffd506c4bb007527a44a1_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/c444781b8425baeb28512d38a88bdedf025c0470c989ec000c441633ca332f60_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/c444781b8425baeb28512d38a88bdedf025c0470c989ec000c441633ca332f60_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/57e5b863e9a8f8ea55c6f9f56e795bee9ecd206dbf0811a73f884c27361c35d4_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/57e5b863e9a8f8ea55c6f9f56e795bee9ecd206dbf0811a73f884c27361c35d4_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; Ściana boczna jest trójkątem równoramiennym. Oznaczmy&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/78f8552c588b1008b51adfd0255ee7dbb85e4c1c9babbcec3cf615072a5e69f5_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/78f8552c588b1008b51adfd0255ee7dbb85e4c1c9babbcec3cf615072a5e69f5_big.svg" type="image/svg+xml"></object> &nbsp;- wysokość ściany bocznej, a <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2e7d2c03a9507ae265ecf5b5356885a53393a2029d241394997265a1a25aefc6_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2e7d2c03a9507ae265ecf5b5356885a53393a2029d241394997265a1a25aefc6_big.svg" type="image/svg+xml"></object> &nbsp;- krawędź boczną.&nbsp; Wykorzystajmy wzór&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/1afe5b19e0263bbc09883101feb0f9ab1e390d0e7b8e4a4b6d44089f4d9678dc_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/1afe5b19e0263bbc09883101feb0f9ab1e390d0e7b8e4a4b6d44089f4d9678dc_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b417844a9edb2dcc9f70b9551f09e6f4556c8cbddd27133760711ac8caa836db_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b417844a9edb2dcc9f70b9551f09e6f4556c8cbddd27133760711ac8caa836db_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8beceb1ffb8531e21e91d4665b2a099ec58bb89d4aacd04cc2da18bfff3cecc7_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8beceb1ffb8531e21e91d4665b2a099ec58bb89d4aacd04cc2da18bfff3cecc7_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/11fc9336756014912b484439e34a35093235c4635c4ec3b18a8ab275e13f968a_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/11fc9336756014912b484439e34a35093235c4635c4ec3b18a8ab275e13f968a_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/485e3dc69c8be8fe35fbc05b6a65438f4f94f78d3a3a1bce02bbe197536ee661_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/485e3dc69c8be8fe35fbc05b6a65438f4f94f78d3a3a1bce02bbe197536ee661_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/732b61134a98f1df4a6fe1c7f8846290eab16e20bb0ed9de4ad30006901ecd41_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/732b61134a98f1df4a6fe1c7f8846290eab16e20bb0ed9de4ad30006901ecd41_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/5289acaf9b0c6b019fc4668a59dee01f81bdd1661fa2806676a8452afff826a8_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/5289acaf9b0c6b019fc4668a59dee01f81bdd1661fa2806676a8452afff826a8_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/0aa115937aae18d1cf450c11160dbb3ae923cf9a8e96ea357829bb168d8da6a9_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/0aa115937aae18d1cf450c11160dbb3ae923cf9a8e96ea357829bb168d8da6a9_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; Zauważmy, że: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/3741bce51fe895f39c41962ec0b1270a5b6e36bb5b304818b672b776c22fbcc1_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/3741bce51fe895f39c41962ec0b1270a5b6e36bb5b304818b672b776c22fbcc1_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/aef62c387d06b1cff37d1f23a6af26009d116f408951a8a4186e13fdf32a7322_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/aef62c387d06b1cff37d1f23a6af26009d116f408951a8a4186e13fdf32a7322_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/13001a731418a145a64415746af4122bfaffb62c880e067311142baf1530b92f_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/13001a731418a145a64415746af4122bfaffb62c880e067311142baf1530b92f_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/4ad72df1b7827f93c33d406fd9402660a5828b560745dd7f41a0c3258f2b8ffb_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/4ad72df1b7827f93c33d406fd9402660a5828b560745dd7f41a0c3258f2b8ffb_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/537a65e7457e6fa8a0c915be2376c4cfe60d2f68731ebf6d7d94d10d9a5e3092_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/537a65e7457e6fa8a0c915be2376c4cfe60d2f68731ebf6d7d94d10d9a5e3092_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/5d7a37106bae3974b0bc340e38d53676ef52ff466affab62ba6549fef46dd4a5_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/5d7a37106bae3974b0bc340e38d53676ef52ff466affab62ba6549fef46dd4a5_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/00674b1d28bc1afdafac55502dfac3ffbf29137c7479d8d386b9678d84755c72_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/00674b1d28bc1afdafac55502dfac3ffbf29137c7479d8d386b9678d84755c72_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/a5361df9a25c2533b827cb470169f8b54dceaff42d1153d95eebc3972eb3db65_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/a5361df9a25c2533b827cb470169f8b54dceaff42d1153d95eebc3972eb3db65_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/68b55b6c240c78307bd9ea5acb1bff1f4782fcede6bed55fd4f823f57aab7ff3_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/68b55b6c240c78307bd9ea5acb1bff1f4782fcede6bed55fd4f823f57aab7ff3_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/17fb3a0921d796ac254c262f3bb782296d5ab1ee2412d96ddf7b229d866e3826_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/17fb3a0921d796ac254c262f3bb782296d5ab1ee2412d96ddf7b229d866e3826_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/426dcaa64dfd183fded4e052983b1537bc8f2021be03779de1761d433d720c00_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/426dcaa64dfd183fded4e052983b1537bc8f2021be03779de1761d433d720c00_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/deb0f0d2e5a7427531c3ae9f7056b4f1055f1e8f110d25553b0cc94cfe8ed258_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/deb0f0d2e5a7427531c3ae9f7056b4f1055f1e8f110d25553b0cc94cfe8ed258_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/e3a5b97706f433175ad4c8ed130119a562b435380d50ad4530bc225b663c6897_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/e3a5b97706f433175ad4c8ed130119a562b435380d50ad4530bc225b663c6897_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2d7d98a6cfd121116418725cc4f150649c09eaaae6241ca28a52dd097d965c1c_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2d7d98a6cfd121116418725cc4f150649c09eaaae6241ca28a52dd097d965c1c_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2ab9b2ec29b40238830b5fa52c3d43092542dfe1082e2bcc6f0cee0002b24d5d_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2ab9b2ec29b40238830b5fa52c3d43092542dfe1082e2bcc6f0cee0002b24d5d_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/812916c986bd087a8c063c7283a91ede1c061429d90efbc0a2c752d971b1f8fb_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/812916c986bd087a8c063c7283a91ede1c061429d90efbc0a2c752d971b1f8fb_big.svg" type="image/svg+xml"></object> &nbsp;

Aby były dwa rozwiązania, to prawa strona równania musi być dodatnia. <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b2acf93f453ace8728e65bfda6198e719bbbab26d5fcc185486e1c2f8360f9c3_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b2acf93f453ace8728e65bfda6198e719bbbab26d5fcc185486e1c2f8360f9c3_big.svg" type="image/svg+xml"></object>

<img class="img-responsive img-thumbnail dropzone-img" src="https://odrabiamy.pl//uploads/exercise_photo/image/104897/2_20159.jpg"> Wektory OC, OA, OB są równej długości, więc&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/5596a2a57e77d267cb1ed976f4ef38089a9b36b1c6acde118cdc5bf3998fe5cb_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/5596a2a57e77d267cb1ed976f4ef38089a9b36b1c6acde118cdc5bf3998fe5cb_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/0e391c5357f13687cc8722208a7f089b26322f5fad126f381d5c9f8814cb5974_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/0e391c5357f13687cc8722208a7f089b26322f5fad126f381d5c9f8814cb5974_big.svg" type="image/svg+xml"></object>

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