https://peda.net/id/e100e9e8ca2 2019-08-29T10:16:11+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/3f8c0526ca3 2019-08-29T11:16:05+03:00 <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(-3%7B%2C%7D4%5Cright)" alt="\left(-3{,}4\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D-%5Cfrac%7B1%7D%7B2%7Dx%2B3%5C%20%5Cfrac%7B1%7D%7B2%7D" alt="y=-\frac{1}{2}x+3\ \frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B1%7D%7B2%7Dx-y%2B3%5C%20%5Cfrac%7B1%7D%7B2%7D%3D0" alt="-\frac{1}{2}x-y+3\ \frac{1}{2}=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft%7C-%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(-3%5Cright)%2B%5Cleft(-1%5Cright)%5Ccdot4%2B3%5C%20%5Cfrac%7B1%7D%7B2%7D%5Cright%7C%7D%7B%5Csqrt%7B%5Cleft(-%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E2%2B%5Cleft(-1%5Cright)%5E2%7D%7D%3D%5Cfrac%7B1%7D%7B%5Cfrac%7B%5Csqrt%7B5%7D%7D%7B2%7D%7D%3D%5Cfrac%7B2%5Csqrt%7B5%7D%7D%7B5%7D" alt="\frac{\left|-\frac{1}{2}\cdot\left(-3\right)+\left(-1\right)\cdot4+3\ \frac{1}{2}\right|}{\sqrt{\left(-\frac{1}{2}\right)^2+\left(-1\right)^2}}=\frac{1}{\frac{\sqrt{5}}{2}}=\frac{2\sqrt{5}}{5}"/> 2019-08-29T11:16:05+03:00 https://peda.net/id/2bff3024ca2 2019-08-29T11:08:08+03:00 <div>Tehtävä 2<br/> Suoran yhtälö on <img src="https://math-demo.abitti.fi/math.svg?latex=2x-y%2B3%3D0" alt="2x-y+3=0"/> . Määritä ilman teknisiä apuvälineitä suoran normaalin yhtälö, joka kulkee pisteen (0, 3) kautta.</div> <div><br/> <div>Suoran yhtälö on <img src="https://math-demo.abitti.fi/math.svg?latex=2x-y%2B3%3D0" alt="2x-y+3=0"/> . Määritä ilman teknisiä apuvälineitä suoran normaalin yhtälö, joka kulkee pisteen (0, 3) kautta.</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=kulma%5Cker%20roin%5C%20suoralle" alt="kulma\ker roin\ suoralle"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%3D%5Cfrac%7B-1%7D%7B2%7D%3D-%5Cfrac%7B1%7D%7B2%7D" alt="k=\frac{\Delta y}{\Delta x}=\frac{-1}{2}=-\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-%5Cfrac%7B1%7D%7B2%7D%5Ccdot%20x%3D-1" alt="-\frac{1}{2}\cdot x=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D2" alt="x=2"/></div> <div>normaalin kulmakerroin on 2 ja se kulkee pisteen (0, 3) kautta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=y-3%3D2%5Cleft(x-0%5Cright)" alt="y-3=2\left(x-0\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y-3%3D2x" alt="y-3=2x"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D2x-3" alt="y=2x-3"/></div> <br/> Tehtävä 3<br/> Laske pisteen (2, -1) etäisyys suorasta <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D%5Cfrac%7B1%7D%7B2%7Dx-3" alt="y=\frac{1}{2}x-3"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft%7Cax_0%2Bby_0%2Bc%5Cright%7C%7D%7B%5Csqrt%7Ba%5E2%2Bb%5E2%7D%7D" alt="\frac{\left|ax_0+by_0+c\right|}{\sqrt{a^2+b^2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft%7C%5Cfrac%7B1%7D%7B2%7D%5Ccdot2%2B1%5Ccdot%5Cleft(-1%5Cright)-3%5Cright%7C%7D%7B%5Csqrt%7B%5Cleft(%5Cfrac%7B1%7D%7B2%7D%5Cright)%5E2%2B1%5E2%7D%7D%3D%5Cfrac%7B%5Cleft%7C-3%5Cright%7C%7D%7B%5Csqrt%7B%5Cfrac%7B5%7D%7B4%7D%7D%7D%3D%5Cfrac%7B6%5Csqrt%7B5%7D%7D%7B5%7D" alt="\frac{\left|\frac{1}{2}\cdot2+1\cdot\left(-1\right)-3\right|}{\sqrt{\left(\frac{1}{2}\right)^2+1^2}}=\frac{\left|-3\right|}{\sqrt{\frac{5}{4}}}=\frac{6\sqrt{5}}{5}"/></div> <br/> <br/> 378<br/> ympyrän keskipiste P on janojen AC ja BC normaalien leikkauspiste<br/> AC normaali:<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=AC%5C%20keskipiste%3A" alt="AC\ keskipiste:"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B4%2B2%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B5%2B1%7D%7B2%7D%5Cright)%3D%5Cleft(3%7B%2C%7D3%5Cright)" alt="\left(\frac{4+2}{2}{,}\ \frac{5+1}{2}\right)=\left(3{,}3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=lasketaan%5C%20janan%5C%20AC%5C%20suuntaisen%5C%20suoran%5C%20kulma%5Cker%20roin" alt="lasketaan\ janan\ AC\ suuntaisen\ suoran\ kulma\ker roin"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%3D%5Cfrac%7B5-1%7D%7B4-2%7D%3D%5Cfrac%7B4%7D%7B2%7D%3D2" alt="k=\frac{\Delta y}{\Delta x}=\frac{5-1}{4-2}=\frac{4}{2}=2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=janan%5C%20AC%5C%20kulma%5Cker%20toimen%5C%20ja%5C%20normaalin%5C%20tulo%5C%20on%5C%20-1" alt="janan\ AC\ kulma\ker toimen\ ja\ normaalin\ tulo\ on\ -1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=2x%3D-1" alt="2x=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D-%5Cfrac%7B1%7D%7B2%7D" alt="x=-\frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y-y_0%3Dk%5Cleft(x-x_0%5Cright)" alt="y-y_0=k\left(x-x_0\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y-3%3Dk%5Cleft(x-3%5Cright)" alt="y-3=k\left(x-3\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3D-%5Cfrac%7B1%7D%7B2%7Dx%2B4%5C%20%5Cfrac%7B1%7D%7B2%7D" alt="y=-\frac{1}{2}x+4\ \frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=janan%5C%20BC%5C%20keskipiste%3A" alt="janan\ BC\ keskipiste:"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7B4%2B7%7D%7B2%7D%7B%2C%7D%5C%20%5Cfrac%7B5%2B2%7D%7B2%7D%5Cright)%3D%5Cleft(5%5C%20%5Cfrac%7B1%7D%7B2%7D%7B%2C%7D%5C%203%5C%20%5Cfrac%7B1%7D%7B2%7D%5Cright)" alt="\left(\frac{4+7}{2}{,}\ \frac{5+2}{2}\right)=\left(5\ \frac{1}{2}{,}\ 3\ \frac{1}{2}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=lasketaan%5C%20BC%5C%20suuntaisen%5C%20suoran%5C%20kulma%5Cker%20roin" alt="lasketaan\ BC\ suuntaisen\ suoran\ kulma\ker roin"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7B%5CDelta%20y%7D%7B%5CDelta%20x%7D%3D%5Cfrac%7B5-2%7D%7B4-7%7D%3D-%5Cfrac%7B3%7D%7B3%7D%3D-1" alt="k=\frac{\Delta y}{\Delta x}=\frac{5-2}{4-7}=-\frac{3}{3}=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=janan%5C%20BC%5C%20kulma%5Cker%20toimen%5C%20ja%5C%20normaalin%5C%20tulo%5C%20on%5C%20-1" alt="janan\ BC\ kulma\ker toimen\ ja\ normaalin\ tulo\ on\ -1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=-1x%3D-1" alt="-1x=-1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1" alt="x=1"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y-y_0%3Dk%5Cleft(x-x_0%5Cright)" alt="y-y_0=k\left(x-x_0\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y-3%5C%20%5Cfrac%7B1%7D%7B2%7D%3Dx-5%5C%20%5Cfrac%7B1%7D%7B2%7D" alt="y-3\ \frac{1}{2}=x-5\ \frac{1}{2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=y%3Dx%2B2" alt="y=x+2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=suorien%5C%20y%3D-%5Cfrac%7B1%7D%7B2%7Dx%2B4%5C%20%5Cfrac%7B1%7D%7B2%7D%5C%20ja%5C%20y%3Dx%2B2%5C%20leikkauspiste" alt="suorien\ y=-\frac{1}{2}x+4\ \frac{1}{2}\ ja\ y=x+2\ leikkauspiste"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cbegin%7Bcases%7D%0A-%5Cfrac%7B1%7D%7B2%7Dx-y%2B4%5C%20%5Cfrac%7B1%7D%7B2%7D%3D0%26%5C%5C%0Ax-y%2B2%3D0%26%0A%5Cend%7Bcases%7D" alt="\begin{cases}&#10;-\frac{1}{2}x-y+4\ \frac{1}{2}=0&amp;\\&#10;x-y+2=0&amp;&#10;\end{cases}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=ratkaistaan%5C%20laskimella" alt="ratkaistaan\ laskimella"/><!--filtered attribute: style="display: inline;"--><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D1%5C%20%5Cfrac%7B2%7D%7B3%7D%7B%2C%7D%5C%20y%3D3%5C%20%5Cfrac%7B2%7D%7B3%7D" alt="x=1\ \frac{2}{3}{,}\ y=3\ \frac{2}{3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=ympyr%C3%A4n%5C%20keskipiste%5C%20P%5C%20on%5C%20%5Cleft(1%5C%20%5Cfrac%7B2%7D%7B3%7D%7B%2C%7D%5C%204%5C%20%5Cfrac%7B2%7D%7B3%7D%5Cright)" alt="ympyrän\ keskipiste\ P\ on\ \left(1\ \frac{2}{3}{,}\ 4\ \frac{2}{3}\right)"/><br/> 380<br/> jotta piste olisi x-akselilla, sen y-koordinaatin on oltava 0<br/> suoran yhtälö x-3y+4<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x-3y%2B4%3D0" alt="x-3y+4=0"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=pisteen%5C%20et%C3%A4isyys%5C%20suorasta%5C%20saadaan%5C%20kaavalla" alt="pisteen\ etäisyys\ suorasta\ saadaan\ kaavalla"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft%7Cax_0%2Bby_0%2Bc%5Cright%7C%7D%7B%5Csqrt%7Ba%5E2%2Bb%5E2%7D%7D" alt="\frac{\left|ax_0+by_0+c\right|}{\sqrt{a^2+b^2}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=sijoitetaan%5C%20lukuja%5C%20kaavaan" alt="sijoitetaan\ lukuja\ kaavaan"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft%7C1x%2B%5Cleft(-3%5Cright)0%2B4%5Cright%7C%7D%7B%5Csqrt%7B1%5E2%2B%5Cleft(-3%5Cright)%5E2%7D%7D%3D%5Csqrt%7B10%7D" alt="\frac{\left|1x+\left(-3\right)0+4\right|}{\sqrt{1^2+\left(-3\right)^2}}=\sqrt{10}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft%7Cx%2B4%5Cright%7C%7D%7B%5Csqrt%7B10%7D%7D%3D%5Csqrt%7B10%7D%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Ccdot%5Csqrt%7B10%7D%5Cright%7C" alt="\frac{\left|x+4\right|}{\sqrt{10}}=\sqrt{10}\ \ \ \ \left|\cdot\sqrt{10}\right|"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%7Cx%2B4%5Cright%7C%3D10" alt="\left|x+4\right|=10"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D6%5C%20tai%5C%20-14" alt="x=6\ tai\ -14"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=pisteet%5C%20ovat%5C%20%5Cleft(6%7B%2C%7D%5C%200%5Cright)%5C%20ja%5C%20%5Cleft(-14%7B%2C%7D%5C%200%5Cright)" alt="pisteet\ ovat\ \left(6{,}\ 0\right)\ ja\ \left(-14{,}\ 0\right)"/><br/> <br/> <br/> Muita hyviä tehtäviä: 372, 375, 376, 377, 383, 387, 388 ja 389.</div> 2019-08-29T10:32:36+03:00