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MATeMAtyka 2. Zakres podstawowy. Reforma 2019

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Oznaczmy: α &nbsp; - miara kąta między bokami&nbsp; a &nbsp; i&nbsp; b &nbsp; <hr> a) &nbsp; P=21​absinα &nbsp; 12=21​⋅4⋅10⋅sinα &nbsp; &nbsp; &nbsp;

Rysunek pomocniczy: <img class="img-responsive img-thumbnail dropzone-img" src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/184812/NR4LsSqWU%2BUbSm7sSvf5Iw%3D%3D_52d839bebdac4e6e60190a56c529e614b0ed26bdc30b4aea1efa8e765a83f2c7.jpg" width="211" height="151"> ∣∢DAC∣=∣∢CAB∣=30o &nbsp; &nbsp; b2=62+(83<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
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H400000v40H845.2724
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​)2−2⋅6⋅83<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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H400000v40H845.2724
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​⋅cos30o &nbsp; b2=36+64⋅3−963<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
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H400000v40H845.2724
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​⋅23<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​​

Z treści zadania wiemy, że |CD|=|CB|, zatem trójkąt BCD to trójkąt równoramienny. Zatem mamy: ∣∢DBC∣=∣∢BDC∣=δ &nbsp; &nbsp; &nbsp;&nbsp;

a) &nbsp; a2=b2+c2−2bc⋅cosα &nbsp; (213<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​)2=b2+82−2⋅b⋅8⋅cos60∘ &nbsp;&nbsp; 52=b2+64−16b⋅21​ &nbsp; &nbsp; &nbsp;

Przypomnijmy: Największy kąt leży naprzeciwko najdłuższego boku. <hr> a) &nbsp; b2=a2+c2−2ac⋅cosβ &nbsp; (23<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
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H400000v40H845.2724
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c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​)2=32+22−2⋅3⋅2⋅cosβ &nbsp; &nbsp;&nbsp;

Oznaczmy: x &nbsp; - długość krótszego boku równoległoboku 3x &nbsp; - długość dłuższego boku równoległoboku&nbsp; Z treści zadania wiemy, że: 2⋅x+2⋅3x=16 [cm] &nbsp; &nbsp; &nbsp;

Oznaczmy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8ed3f6ad685b959ead7022518e1af76cd816f8e8ec7ccdda1ed4018e8f2223f8_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8ed3f6ad685b959ead7022518e1af76cd816f8e8ec7ccdda1ed4018e8f2223f8_big.svg" type="image/svg+xml"></object> &nbsp; - miara kąta między bokami&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/ca978112ca1bbdcafac231b39a23dc4da786eff8147c4e72b9807785afee48bb_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/ca978112ca1bbdcafac231b39a23dc4da786eff8147c4e72b9807785afee48bb_big.svg" type="image/svg+xml"></object> &nbsp; i&nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/3e23e8160039594a33894f6564e1b1348bbd7a0088d42c4acb73eeaed59c009d_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/3e23e8160039594a33894f6564e1b1348bbd7a0088d42c4acb73eeaed59c009d_big.svg" type="image/svg+xml"></object> &nbsp; <hr> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8075922a98308b039ef4cde06ff146e47c8def0a9f3d24f195b2b004d14dac3f_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8075922a98308b039ef4cde06ff146e47c8def0a9f3d24f195b2b004d14dac3f_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/da60000e097941de85273ed5142d84ec8e3fddeabed11d54e91bed7dc9cfa400_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/da60000e097941de85273ed5142d84ec8e3fddeabed11d54e91bed7dc9cfa400_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2e2ec4bddbdcde4e44d6dc5264f1c0e6f2a74197d51bb84602245f0ae876015d_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2e2ec4bddbdcde4e44d6dc5264f1c0e6f2a74197d51bb84602245f0ae876015d_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; &nbsp;

Rysunek pomocniczy: <img class="img-responsive img-thumbnail dropzone-img" src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/184812/NR4LsSqWU%2BUbSm7sSvf5Iw%3D%3D_52d839bebdac4e6e60190a56c529e614b0ed26bdc30b4aea1efa8e765a83f2c7.jpg" width="211" height="151"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/66fa43264bbd402dbb8c075fe6d36c453aa58889cfe57a7c4a842503b8a1c368_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/66fa43264bbd402dbb8c075fe6d36c453aa58889cfe57a7c4a842503b8a1c368_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b5dd2e63c0cb6267f95da4719ea33cfbd47649b3c3a4f38021bf6ca786736f94_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b5dd2e63c0cb6267f95da4719ea33cfbd47649b3c3a4f38021bf6ca786736f94_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2055796c11838f472988c81ee160f157d135c0af25d70d943d2a4d0b9e9f2263_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2055796c11838f472988c81ee160f157d135c0af25d70d943d2a4d0b9e9f2263_big.svg" type="image/svg+xml"></object>

Rysunek pomocniczy: <img src="https://prod.odrabiamy-assets.pl/uploads/content_image/image/126377/v0nkGo3NAzzLHhAI6RLYlQ%3D%3D_b60abf05f8ca458361bfc2d56e2d746d7149e88843048fe87d8a821d213d9e32.jpg" alt="Thumb 10 20362"> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/3f218ebcfdc9f9a5ebd9e42b2f2c43339e96bf1731857261f1355de1965d3b02_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/3f218ebcfdc9f9a5ebd9e42b2f2c43339e96bf1731857261f1355de1965d3b02_big.svg" type="image/svg+xml"></object> &nbsp;&nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/143d827f843779dcf37446dc9497c8e75dc968e2bf7af8f85a4aecdea898cd2d_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/143d827f843779dcf37446dc9497c8e75dc968e2bf7af8f85a4aecdea898cd2d_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/ac4270adf87e134e40f0dc4cb8724f0a7ca288078bf0ab59ae90f3b6d694c90e_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/ac4270adf87e134e40f0dc4cb8724f0a7ca288078bf0ab59ae90f3b6d694c90e_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; Ze związków trygonometrycznych w trójkącie ACD otrzymujemy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b6205f162ca0644d71318c2eb1aa80d7951e9b9caad749d6f8131d7b800a7cba_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b6205f162ca0644d71318c2eb1aa80d7951e9b9caad749d6f8131d7b800a7cba_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/9aaa19df5a7b8fdcd197aaf063adcb8cfbac53aba0b6d5ba889e77203b3e390a_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/9aaa19df5a7b8fdcd197aaf063adcb8cfbac53aba0b6d5ba889e77203b3e390a_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/cc59702dda1b3b6b09dd327dab82e85030452a1077427c5e33d5e11a52f7dbe1_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/cc59702dda1b3b6b09dd327dab82e85030452a1077427c5e33d5e11a52f7dbe1_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/91b67e493257dcbbca40906af1672709a5adeed53ec00c1c41cb58d7c31cbc39_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/91b67e493257dcbbca40906af1672709a5adeed53ec00c1c41cb58d7c31cbc39_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/919d3352814424e6e08eac562aa04281ad655d57fa1518dc2526dd9ae718d6ab_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/919d3352814424e6e08eac562aa04281ad655d57fa1518dc2526dd9ae718d6ab_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/dfb3c48cd4fd8669b0bd62aa37849e2fe830c4436937f45ba7756189cfdc769e_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/dfb3c48cd4fd8669b0bd62aa37849e2fe830c4436937f45ba7756189cfdc769e_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/6e2aebbe5c54b8b180fd92183a97808c07ff250c90c8567b93476485f8ccb4e6_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/6e2aebbe5c54b8b180fd92183a97808c07ff250c90c8567b93476485f8ccb4e6_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/fe5c201cffc096a44dc64f2dcf7502b2d135efb5e4a3d78eb20201cdee67b9f2_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/fe5c201cffc096a44dc64f2dcf7502b2d135efb5e4a3d78eb20201cdee67b9f2_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; Z tw. Pitagorasa w trójkącie ACP otrzymujemy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/3b41d44262040cb7578d3c449f4ec39e93ef3bc8efb5285b3f94ffde3e3f9087_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/3b41d44262040cb7578d3c449f4ec39e93ef3bc8efb5285b3f94ffde3e3f9087_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/c4d1001cf3a4869d68f3c9db93f3f62d05a2c5714f0db5e01c9bc1c6632e3a9c_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/c4d1001cf3a4869d68f3c9db93f3f62d05a2c5714f0db5e01c9bc1c6632e3a9c_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/ef11d0b4fc7964d462460b95cc00d3731d4ecf955ec6991dd860a68d8c76e279_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/ef11d0b4fc7964d462460b95cc00d3731d4ecf955ec6991dd860a68d8c76e279_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/f6fd4c8cbe21d698583c6675e818c8ab01a4b527df3aeef85b2703131ac2d611_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/f6fd4c8cbe21d698583c6675e818c8ab01a4b527df3aeef85b2703131ac2d611_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/befdce52eb840062c7d1f215a47e52eb2ade43f9da1290d699d9811a44954def_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/befdce52eb840062c7d1f215a47e52eb2ade43f9da1290d699d9811a44954def_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; Z tw. Pitagorasa w trójkącie BEP otrzymujemy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/3679702725c02895c96de68fed026f97228b9b88be849bd6dc054f616f056704_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/3679702725c02895c96de68fed026f97228b9b88be849bd6dc054f616f056704_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/281e3e4989a9f8fee025258d9e4285e849a74ddb6ac77490bfc2686cf70f298e_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/281e3e4989a9f8fee025258d9e4285e849a74ddb6ac77490bfc2686cf70f298e_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/0edb9d4cb8d6b4cacfc0a01b3c78707c76599b9f5210895248175dd9813b653d_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/0edb9d4cb8d6b4cacfc0a01b3c78707c76599b9f5210895248175dd9813b653d_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b3c3ba5f4ab5bcfa00a11e95629bf8d33f77049f22aa813ef094b7ef4870701e_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b3c3ba5f4ab5bcfa00a11e95629bf8d33f77049f22aa813ef094b7ef4870701e_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/4f0a16232fb35a3b957ebca39f38e61c75d97f5f08ffa1f880c1fb028d15af98_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/4f0a16232fb35a3b957ebca39f38e61c75d97f5f08ffa1f880c1fb028d15af98_big.svg" type="image/svg+xml"></object> &nbsp;

Z treści zadania wiemy, że |CD|=|CB|, zatem trójkąt BCD to trójkąt równoramienny. Zatem mamy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/be091c24b3cf4f70e465beb25920ad8326874ad13da367f587118de8337ca16e_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/be091c24b3cf4f70e465beb25920ad8326874ad13da367f587118de8337ca16e_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; &nbsp;&nbsp;

Przypomnijmy: Największy kąt leży naprzeciwko najdłuższego boku. <hr> <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/8075922a98308b039ef4cde06ff146e47c8def0a9f3d24f195b2b004d14dac3f_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/8075922a98308b039ef4cde06ff146e47c8def0a9f3d24f195b2b004d14dac3f_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/371bd47bb326da8163ee8dafc979f79d9c9dfd642304edbe8a4a07c4347ca044_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/371bd47bb326da8163ee8dafc979f79d9c9dfd642304edbe8a4a07c4347ca044_big.svg" type="image/svg+xml"></object> &nbsp; <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b490281eee45be706057909584c0ded566ffdb5c6cfb696754724151c8e841cc_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b490281eee45be706057909584c0ded566ffdb5c6cfb696754724151c8e841cc_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp;&nbsp;

Oznaczmy: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/2d711642b726b04401627ca9fbac32f5c8530fb1903cc4db02258717921a4881_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/2d711642b726b04401627ca9fbac32f5c8530fb1903cc4db02258717921a4881_big.svg" type="image/svg+xml"></object> &nbsp; - długość krótszego boku równoległoboku <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/35e93eb65d07f8e8fd5910c5ebe5925822a5d856f4b1c3c1af6e9f2b9193b806_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/35e93eb65d07f8e8fd5910c5ebe5925822a5d856f4b1c3c1af6e9f2b9193b806_big.svg" type="image/svg+xml"></object> &nbsp; - długość dłuższego boku równoległoboku&nbsp; Z treści zadania wiemy, że: <object class="math small" data="https://odrabiamy.pl/api/v3/math_formulas/b7b140cdda01913d3f40eddb5dd9e6d94ec654b6953a3f083f02325dba8827f1_small.svg" type="image/svg+xml"></object><object class="math big" data="https://odrabiamy.pl/api/v3/math_formulas/b7b140cdda01913d3f40eddb5dd9e6d94ec654b6953a3f083f02325dba8827f1_big.svg" type="image/svg+xml"></object> &nbsp; &nbsp; &nbsp;

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