https://peda.net/id/0c30327c71e 2025-08-05T13:50:46+03:00 https://peda.net/:static/535/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/0c3521bc71e 2018-04-25T09:36:18+03:00 235 a)<br/> <div> </div> <div>Piste F on janan BC keskipiste:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cleft(%5Cfrac%7B(-1%2B6)%7D%7B2%7D%7B%2C%7D%5Cfrac%7B(5%2B0)%7D%7B2%7D%5Cright)%3D%5Cleft(%5Cfrac%7B5%7D%7B2%7D%7B%2C%7D%5Cfrac%7B5%7D%7B2%7D%5Cright)" alt="F=\left(\frac{(-1+6)}{2}{,}\frac{(5+0)}{2}\right)=\left(\frac{5}{2}{,}\frac{5}{2}\right)"/></div> <br/> <div>AP = 2/3 * AF (mediaanien leikkauspiste jakaa mediaanit suhteessa 2 : 1 kärjestä lukien)</div> <div> </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOP%7D%3D%5Coverline%7BOA%7D%2B%5Coverline%7BAP%7D%3D%5Coverline%7BOA%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Coverline%7BAF%7D" alt="\overline{OP}=\overline{OA}+\overline{AP}=\overline{OA}+\frac{2}{3}\overline{AF}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2i%2B2j%2B%5Cfrac%7B2%7D%7B3%7D%5Cleft(%5Cleft(%5Cfrac%7B5%7D%7B2%7D-%5Cleft(-%5Cfrac%7B4%7D%7B2%7D%5Cright)%5Cright)i%2B%5Cfrac%7B1%7D%7B2%7Dj%5Cright)" alt="=-2i+2j+\frac{2}{3}\left(\left(\frac{5}{2}-\left(-\frac{4}{2}\right)\right)i+\frac{1}{2}j\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2i%2B2j%2B%5Cfrac%7B2%7D%7B3%7D%5Cleft(%5Cfrac%7B9%7D%7B2%7Di%2B%5Cfrac%7B1%7D%7B2%7Dj%5Cright)" alt="=-2i+2j+\frac{2}{3}\left(\frac{9}{2}i+\frac{1}{2}j\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2i%2B2j%2B%5Cfrac%7B2%7D%7B3%7D%5Ccdot%5Cfrac%7B9%7D%7B2%7Di%2B%5Cfrac%7B2%7D%7B3%7D%5Ccdot%5Cfrac%7B1%7D%7B2%7Dj" alt="=-2i+2j+\frac{2}{3}\cdot\frac{9}{2}i+\frac{2}{3}\cdot\frac{1}{2}j"/><br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2i%2B2j%2B3i%2B%5Cfrac%7B1%7D%7B3%7Dj" alt="=-2i+2j+3i+\frac{1}{3}j"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D1i%2B%5Cfrac%7B7%7D%7B3%7Dj" alt="=1i+\frac{7}{3}j"/><br/> <br/> <span class="editor underline">Siis piste P = (1,</span> 7/3)<br/> <br/> <b>Toinen reitti: AB + ½BC:<br/> </b> <div> <div><b>OP = OA + AP</b><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Coverline%7BOA%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Cleft(%5Coverline%7BAB%7D%2B%5Cfrac%7B1%7D%7B2%7D%5Coverline%7BBC%7D%5Cright)" alt="=\overline{OA}+\frac{2}{3}\left(\overline{AB}+\frac{1}{2}\overline{BC}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2%5Coverline%7Bi%7D%2B2%5Coverline%7Bj%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Cleft(%5Cleft(%5Cleft(6-%5Cleft(-2%5Cright)%5Cright)%5Coverline%7Bi%7D%2B%5Cleft(0-2%5Cright)%5Coverline%7Bj%7D%5Cright)%2B%5Cfrac%7B1%7D%7B2%7D%5Cleft(-7%5Coverline%7Bi%7D%2B5%5Coverline%7Bj%7D%5Cright)%5Cright)" alt="=-2\overline{i}+2\overline{j}+\frac{2}{3}\left(\left(\left(6-\left(-2\right)\right)\overline{i}+\left(0-2\right)\overline{j}\right)+\frac{1}{2}\left(-7\overline{i}+5\overline{j}\right)\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2%5Coverline%7Bi%7D%2B2%5Coverline%7Bj%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Cleft(8%5Coverline%7Bi%7D-2%5Coverline%7Bj%7D-%5Cfrac%7B7%7D%7B2%7Di%2B%5Cfrac%7B5%7D%7B2%7D%5Coverline%7Bj%7D%5Cright)" alt="=-2\overline{i}+2\overline{j}+\frac{2}{3}\left(8\overline{i}-2\overline{j}-\frac{7}{2}i+\frac{5}{2}\overline{j}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-2%5Coverline%7Bi%7D%2B2%5Coverline%7Bj%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Ccdot8%5Coverline%7Bi%7D-%5Cfrac%7B2%7D%7B3%7D%5Ccdot2%5Coverline%7Bj%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Ccdot%5Cfrac%7B-7%7D%7B2%7D%5Coverline%7Bi%7D%2B%5Cfrac%7B2%7D%7B3%7D%5Ccdot%5Cfrac%7B5%7D%7B2%7D%5Coverline%7Bj%7D" alt="=-2\overline{i}+2\overline{j}+\frac{2}{3}\cdot8\overline{i}-\frac{2}{3}\cdot2\overline{j}+\frac{2}{3}\cdot\frac{-7}{2}\overline{i}+\frac{2}{3}\cdot\frac{5}{2}\overline{j}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cfrac%7B6%7D%7B3%7D%5Coverline%7Bi%7D%2B%5Cfrac%7B6%7D%7B3%7D%5Coverline%7Bj%7D%2B%5Cfrac%7B16%7D%7B3%7D%5Coverline%7Bi%7D-%5Cfrac%7B4%7D%7B3%7D%5Coverline%7Bj%7D-%5Cfrac%7B7%7D%7B3%7D%5Coverline%7Bi%7D%2B%5Cfrac%7B5%7D%7B3%7D%5Coverline%7Bj%7D" alt="=-\frac{6}{3}\overline{i}+\frac{6}{3}\overline{j}+\frac{16}{3}\overline{i}-\frac{4}{3}\overline{j}-\frac{7}{3}\overline{i}+\frac{5}{3}\overline{j}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cfrac%7B6%7D%7B3%7D%5Coverline%7Bi%7D%2B%5Cfrac%7B16%7D%7B3%7D%5Coverline%7Bi%7D-%5Cfrac%7B7%7D%7B3%7D%5Coverline%7Bi%7D%2B%5Cfrac%7B6%7D%7B3%7D%5Coverline%7Bj%7D-%5Cfrac%7B4%7D%7B3%7D%5Coverline%7Bj%7D%2B%5Cfrac%7B5%7D%7B3%7D%5Coverline%7Bj%7D" alt="=-\frac{6}{3}\overline{i}+\frac{16}{3}\overline{i}-\frac{7}{3}\overline{i}+\frac{6}{3}\overline{j}-\frac{4}{3}\overline{j}+\frac{5}{3}\overline{j}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cfrac%7B3%7D%7B3%7D%5Coverline%7Bi%7D%2B%5Cfrac%7B7%7D%7B3%7D" alt="=\frac{3}{3}\overline{i}+\frac{7}{3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D1%5Coverline%7Bi%7D%2B%5Cfrac%7B7%7D%7B3%7D%5Coverline%7Bj%7D" alt="=1\overline{i}+\frac{7}{3}\overline{j}"/><br/> <br/> <br/> <b>217</b><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%3D%5Cleft(r-1%5Cright)%5Coverline%7Bi%7D%2B2%5Coverline%7Bj%7D" alt="\overline{u}=\left(r-1\right)\overline{i}+2\overline{j}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bb%7D%3D2r%5Coverline%7Bi%7D%2Br%5Coverline%7Bj%7D" alt="\overline{b}=2r\overline{i}+r\overline{j}"/> <div> </div> <div>Vektorin<span> </span><b>u</b> ja<span> </span><b>b</b> ovat yhdensuuntaisia, kun:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%3Ds%5Coverline%7Bb%7D" alt="\overline{u}=s\overline{b}"/></div> ja vastakkaissuuntaisia, jos s &lt; 0. <div><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(r-1%5Cright)%5Coverline%7Bi%7D%2B2%5Coverline%7Bj%7D%3Ds%5Cleft(2r%5Coverline%7Bi%7D%2Br%5Coverline%7Bj%7D%5Cright)" alt="\left(r-1\right)\overline{i}+2\overline{j}=s\left(2r\overline{i}+r\overline{j}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(r-1%5Cright)%5Coverline%7Bi%7D%2B2%5Coverline%7Bj%7D%3Ds%5Ccdot2r%5Coverline%7Bi%7D%2Bsr%5Coverline%7Bj%7D" alt="\left(r-1\right)\overline{i}+2\overline{j}=s\cdot2r\overline{i}+sr\overline{j}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0Ar-1%26%3D%262sr%5C%5C%0A2%26%3D%26sr%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;r-1&amp;=&amp;2sr\\&#10;2&amp;=&amp;sr&#10;\end{cases}"/> <div>Alemmasta saadaan <img src="https://math-demo.abitti.fi/math.svg?latex=r%3D%5Cfrac%7B2%7D%7Bs%7D" alt="r=\frac{2}{s}"/>, sijoitetaan ylempään:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7Bs%7D-1%3D2%5Ccdot%20s%5Ccdot%5Cfrac%7B2%7D%7Bs%7D" alt="\frac{2}{s}-1=2\cdot s\cdot\frac{2}{s}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7Bs%7D-1%3D4" alt="\frac{2}{s}-1=4"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B2%7D%7Bs%7D%3D%5Cfrac%7B5%7D%7B1%7D" alt="\frac{2}{s}=\frac{5}{1}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=5s%3D2" alt="5s=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=s%3D%5Cfrac%7B2%7D%7B5%7D%3E0" alt="s=\frac{2}{5}&gt;0"/><!--filtered attribute: style="max-width: 100%; max-height: 1000px; vertical-align: middle; margin: 4px; padding: 3px 10px; cursor: pointer; border: 1px solid #e6f2f8; background: #edf9ff;"--></div> <div>Koska s &gt; 0, vektorit ovat yhdensuuntaiset ja samansuuntaiset.</div> <div> </div> <div><span class="editor underline">V: Ei voi.</span></div> <div><br/> <div> <div><b>218</b><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%3Dk%5Coverline%7Bi%7D-4%5Coverline%7Bj%7D" alt="\overline{u}=k\overline{i}-4\overline{j}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bv%7D%3D-9%5Coverline%7Bi%7D%2Bk%5Coverline%7Bj%7D" alt="\overline{v}=-9\overline{i}+k\overline{j}"/><br/> <div>Vektorit u ja v ovat yhdensuuntaiset, jos:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bu%7D%3Ds%5Coverline%7Bv%7D" alt="\overline{u}=s\overline{v}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=..." alt="..."/><br/> <div>yhtälöpari</div> <div>ratkaistaan k</div> <br/> <b>219</b><br/> vektorit AC, BD tiedossa<br/> sivut ovat vektorit AD, AB, muodostetaan<br/> nyt ala = |AD|*|AB|<br/> <br/> <b>239</b><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOC%7D%3D%5Coverline%7BOA%7D%2B20%5Ccdot%5Coverline%7Bu%7D%5E0%2B4%5Ccdot%5Coverline%7Bv%7D%5E0" alt="\overline{OC}=\overline{OA}+20\cdot\overline{u}^0+4\cdot\overline{v}^0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BOA%7D%3D%5Coverline%7BOC%7D-20%5Ccdot%5Coverline%7Bu%7D%5E0-4%5Ccdot%5Coverline%7Bv%7D%5E0" alt="\overline{OA}=\overline{OC}-20\cdot\overline{u}^0-4\cdot\overline{v}^0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Coverline%7BOC%7D-20%5Ccdot%5Cfrac%7B%5Coverline%7Bu%7D%7D%7B%5Cleft%7C%5Coverline%7Bu%7D%5Cright%7C%7D-4%5Ccdot%5Cfrac%7B%5Coverline%7Bv%7D%7D%7B%5Cleft%7C%5Coverline%7Bv%7D%5Cright%7C%7D" alt="=\overline{OC}-20\cdot\frac{\overline{u}}{\left|\overline{u}\right|}-4\cdot\frac{\overline{v}}{\left|\overline{v}\right|}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D%5Cleft(15i%2B4j%5Cright)-20%5Ccdot%5Cfrac%7B3i-%5Cfrac%7B8%7D%7B5%7Dj%7D%7B%5Csqrt%7B3%5E2%2B%5Cleft(%5Cfrac%7B8%7D%7B5%7D%5Cright)%5E2%7D%7D-4%5Ccdot%5Cfrac%7B-%5Cfrac%7B5%7D%7B2%7Di%2B6j%7D%7B%5Csqrt%7B%5Cleft(%5Cfrac%7B5%7D%7B2%7D%5Cright)%5E2%2B6%5E2%7D%7D" alt="=\left(15i+4j\right)-20\cdot\frac{3i-\frac{8}{5}j}{\sqrt{3^2+\left(\frac{8}{5}\right)^2}}-4\cdot\frac{-\frac{5}{2}i+6j}{\sqrt{\left(\frac{5}{2}\right)^2+6^2}}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D-%5Cfrac%7B245%7D%7B221%7Di%2B%5Cfrac%7B2148%7D%7B221%7Dj" alt="=-\frac{245}{221}i+\frac{2148}{221}j"/></div> <div><span>joten</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cleft(-%5Cfrac%7B245%7D%7B221%7D%7B%2C%7D%5Cfrac%7B2148%7D%7B221%7D%5Cright)" alt="A=\left(-\frac{245}{221}{,}\frac{2148}{221}\right)"/></div> <br/> <br/> </div> </div> </div> </div> 2025-08-05T13:50:46+03:00