Teht\u00e4v\u00e4 210. a)<\/div>\n\n\n\n
Teht\u00e4v\u00e4 210. b)<\/div>\n\n\n\n
Koska piste ![\"C\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-80fc70eb30b18f9c8307c5d121c42bf5_l3.png\" "\\"Rendered")
on ympyr\u00e4ll\u00e4, sen et\u00e4isyys pisteest\u00e4 ![\"A\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-07b1ce040c95b146f97c947b6776383f_l3.png\" "\\"Rendered")
on ympyr\u00e4n s\u00e4de.<\/p>\n\n\n\n
Vastaus<\/strong>: Ympyr\u00e4n s\u00e4de on![\"\sqrt{5}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-dbce06882b7d3d518529141d2ccf6e15_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\nTeht\u00e4v\u00e4 213. a)<\/div>\n\n\n\n
![\"\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/uploads\/sites\/136\/2019\/08\/213c.png\")
<\/figure><\/div>\n\n\n\n![\"A = (-2, -1)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-80abcc6b680cb7bdd86577cd57f032be_l3.png\" "\\"Rendered")
, ![\"B = (2, 1)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-0fba3baab73fdee98921018b01d2033c_l3.png\" "\\"Rendered")
ja ![\"C = (-1, 2)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-52f6cc629c53257457c6c2d66c873bd1_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n
Kolmio on tasakylkinen, jos siin\u00e4 kaksi yht\u00e4 pitk\u00e4\u00e4 sivua. Lasketaan kolmion sivujen pituudet.<\/p>\n\n\n\n
Sivun ![\"AC\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-9b49745954302728d493a7bf03860395_l3.png\" "\\"Rendered")
pituus on pisteiden ja et\u00e4isyys:
<\/span> <\/span>![\"\begin{align*}\sqrt{(-1-(-2))^2+(2-(-1))^2}&=\sqrt{(-1+2)^2+(2+1)^2}\\&=\sqrt{(1^2+3^2}\\&=\sqrt{1+9}\\&=\sqrt{10}\end{align*}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-a32a7fbb08a488b9cdabaa6776488f77_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\n Sivun ![\"BC\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-c79871fa121cf86c2623051621ec5f60_l3.png\" "\\"Rendered")
pituus on pisteiden ![\"B\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-a5142ffc089d2df753dd20966a3d4881_l3.png\" "\\"Rendered")
ja et\u00e4isyys.
<\/span> <\/span>![\"\begin{align*}\sqrt{(-1-2))^2+(2-1)^2}&=\sqrt{(-3)^2+1^2}\\&=\sqrt{(9+1}\\&=\sqrt{10}\end{align*}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-be540cdda96691e62589b43c71df3b4d_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nSivun ![\"AB\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-27a4042d131fbdf3df9d9b9353ea3a80_l3.png\" "\\"Rendered")
pituus on pisteiden ja et\u00e4isyys.
<\/span> <\/span>![\"\begin{align*}\sqrt{(2-(-2))^2+(1-(-1))^2}&=\sqrt{(2+2)^2+(1+1)^2}\\&=\sqrt{(4^2+2^2}\\&=\sqrt{16+4}\\&=\sqrt{20}\\&=\sqrt{4 \cdot 5}\\&=2\sqrt{5}\end{align*}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-84611471fca76bfa64ec97f3532818df_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nOikeastaan sivun pituutta ei olisi tarvinnut laskea, koska tasakylkisyyden osoittamiseksi kahden sivun pituuksien tulee olla samat.<\/p>\n\n\n\n
Koska sivut ja ovat yht\u00e4 pitk\u00e4t, kolmio on tasakylkinen. <\/p>\n\n\n\n
213. b)<\/div>\n\n\n\n
Kannan keskipiste on janan keskipiste:<\/p>\n\n\n\n
<\/span> <\/span>![\"\[\left(\frac{-2+2}{2},\frac{-1+1}{2}\right)=(0,0)\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-08340de58819ce3cdba11eeea21a304a_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nTeht\u00e4v\u00e4 213. c)<\/div>\n\n\n\n
Lasketaan kolmion pinta-ala v\u00e4hent\u00e4m\u00e4ll\u00e4 kuvaan piirretyn suorakulmion pinta-alasta kolmen suorakulmaisen kolmion alat. <\/p>\n\n\n\n
Suorakulmion pinta-ala<\/strong>:<\/p>\n\n\n\nSuorakulmion toisen sivun pituus on pisteiden ja ![\"x\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-edf6d5462aa989d30da6f7f7d6e93e0f_l3.png\" "\\"Rendered")
-koordinaattien erotuksen itseisarvo
<\/span> <\/span>![\"\[\left|2 - (-2)\right| = 4\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-1a91bca85337b5618b4f887192ad4cf1_l3.png\" "\\"Rendered")
<\/p> ja toisen sivun pituus on pisteiden ja ![\"y\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-1511b6d339e66309a4071bc95cfd9285_l3.png\" "\\"Rendered")
-koordinaattien erotuksen itseisarvo <\/span> <\/span>![\"\[\left|2 - (-1)\left| = 3\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-87e55bbd9db59fef1736908ebf421157_l3.png\" "\\"Rendered")
<\/p> <\/p>\n\n\n\n Suorakulmion pinta-ala on ![\"4 \cdot 3 = 12\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-faf7a3a869253b91a9897a0e6f097de3_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\n
Suorakulmaisten kolmioiden kateettien pituudet saadaan vastaavasti koordinaattien avulla.
Suurin kolmio alhaalla<\/strong>:
Toinen kateetti on ![\"4\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-1dd00e60174d7692394de525c5126bb4_l3.png\" "\\"Rendered")
ja toinen ![\"\left|1 - (-1)\right| = 2\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-103824b7d9db8584b917efd5ce9bd624_l3.png\" "\\"Rendered")
.
Pinta-ala on <\/span> <\/span>![\"\[\frac{4 \cdot 2}{2}=4\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-c128b7838fae08e2e9fcba9455da8162_l3.png\" "\\"Rendered")
<\/p>
Ylh\u00e4\u00e4ll\u00e4 vasemmalla oleva kolmio<\/strong>:
Toinen kateetti on ![\"3\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-05db02a331d82ee0cac16ca358fd4d43_l3.png\" "\\"Rendered")
ja toinen ![\"\left|-2 - (-1)\left| = 1\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-ccafd65baa4d3aa96e2ede30184fd977_l3.png\" "\\"Rendered")
.
Pinta-ala on <\/span> <\/span>![\"\[\frac{3 \cdot1}{2}=\frac{3}{2}\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-aaecae0988dc0a66c09faa23888f009e_l3.png\" "\\"Rendered")
<\/p>
Ylh\u00e4\u00e4ll\u00e4 oikealla oleva kolmio<\/strong>:
Toinen kateetti on ja toinen ![\"1\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-1fead2774e8b11a333e189478cefe74e_l3.png\" "\\"Rendered")
.
Pinta-ala on <\/span> <\/span>![\"\[\frac{3 \cdot 1}{2}=\frac{3}{2}\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-ca8ae3ea8f25631517762e19e4a2cc7a_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nSiten kolmion pinta-ala on<\/p>\n\n\n\n
<\/span> <\/span>![\"\[12-\left(4+\frac{3}{2} + \frac{3}{2}\right)=12-\left(4+\frac{6}{2}\right)=12-(4+3)=12-7=5\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-2c59bb4981bc5f7a5a68283c49c60543_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nVastaus:<\/strong> Kolmion pinta-ala on ![\"5\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-fc8c173d9e46c0d8030697c2a1d610a9_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\nTeht\u00e4v\u00e4 214. a)<\/div>\n\n\n\n
Muodostetaan k\u00e4yr\u00e4n yht\u00e4l\u00f6 annettujen ehtojen nojalla:<\/p>\n\n\n\n
Pisteen -koordinaatin ja luvun ![\"2\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-00ac69dabbb1414f319584b1b7797538_l3.png\" "\\"Rendered")
erotus on ![\"y - 2\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-b7e34ab088721ee41e0673b9afc33328_l3.png\" "\\"Rendered")
. T\u00e4m\u00e4n neli\u00f6 on ![\"(y - 2)^2\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-590ef7580540950c2739e7371a52b707_l3.png\" "\\"Rendered")
.
-koordinaatin neli\u00f6 on ![\"y^2\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-27a8342ff36d36c2648490e6261bcf82_l3.png\" "\\"Rendered")
. N\u00e4m\u00e4 ovat yht\u00e4 suuret, joten k\u00e4yr\u00e4n yht\u00e4l\u00f6 on
<\/span> <\/span>![\"\[(y - 2)^2 = y^2\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-1b3a946fb0f2593ed9e3518145e2dcc5_l3.png\" "\\"Rendered")
<\/p>. <\/p>\n\n\n\nSievennet\u00e4\u00e4n k\u00e4yr\u00e4n yht\u00e4l\u00f6:<\/p>\n\n\n\n
<\/span> <\/span>![\"\begin{align*}(y - 2)^2 &= y^2\\y^2-2 \cdot y \cdot 2 + (-2)^2&=y^2\\y^2-4y+4&=y^2\\y^2-y^2-4y+4&=0\\-4y&=-4\ \parallel\ :(-4)\\y&=\frac{-4}{-4}\\y&=1\end{align*}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-fca7d9c382224f1e6a3af65bea130943_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nK\u00e4yr\u00e4 on seuraava<\/p>\n\n\n\n
![\"\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/uploads\/sites\/136\/2019\/08\/214b_1.png\")
K\u00e4yr\u00e4n ![\"y=1\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-cf6e1fcc4feb9168c4a942e9d5f66cf7_l3.png\" "\\"Rendered")
kuvaaja.<\/figcaption><\/figure><\/div>\n\n\n\nTeht\u00e4v\u00e4 214. b)<\/div>\n\n\n\n
Pistejoukko koostuu pisteist\u00e4, joiden -koordinaatti on . Olipa -koordinaatti mik\u00e4 tahansa, -koordinaatti on aina . <\/p>\n\n\n\n
Piste ![\"(1,1001)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-2ac3520981398aed732b3e5d6ccb2575_l3.png\" "\\"Rendered")
ei ole<\/strong> k\u00e4yr\u00e4ll\u00e4, koska -koordinaatti ei ole .
Piste ![\"(1001, 1)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-eec6b46b9ef72646cacad205ec30cdd9_l3.png\" "\\"Rendered")
on<\/strong> k\u00e4yr\u00e4ll\u00e4, koska -koordinaatti on .<\/p>\n\n\n\nTeht\u00e4v\u00e4 215. a)<\/div>\n\n\n\n
K\u00e4yr\u00e4n yht\u00e4l\u00f6 on ![\"(x-3)(y+2)=0\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-2251cb1049e1a8c75907a28349fc8665_l3.png\" "\\"Rendered")
. Piiret\u00e4\u00e4n k\u00e4yr\u00e4 GeoGebralla.<\/p>\n\n\n\n
![\"\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/uploads\/sites\/136\/2019\/08\/215a_22.png\")
K\u00e4yr\u00e4 <\/figcaption><\/figure><\/div>\n\n\n\nTeht\u00e4v\u00e4 215. b)<\/div>\n\n\n\n
K\u00e4yr\u00e4ll\u00e4 olevat pisteet toteuttavat yht\u00e4l\u00f6n . <\/p>\n\n\n\n
Tulon nollas\u00e4\u00e4nn\u00f6n perusteella yht\u00e4l\u00f6 toteutuu, kun jompi kumpi tulon tekij\u00f6ist\u00e4 on nolla eli<\/p>\n\n\n\n
<\/span> <\/span>![\"\begin{align*}x-3&=0\\x&=3\end{align*}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-7d6f7e342616a1c25654a79b91802719_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\ntai<\/strong> <\/p>\n\n\n\n
<\/span> <\/span>![\"\begin{align*}y+2&=0\\y&=-2\end{align*}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-18b541765fce7edf49efe3d0fe3c8362_l3.png\" "\\"Rendered")
<\/p> <\/p>\n\n\n\nSiten yht\u00e4l\u00f6 tarkoittaa pistejoukkoa, joka koostuu pisteist\u00e4, joiden
-koodinaatti on 3
ja
-koordinaatti on -2.<\/p>\n\n\n\n
Teht\u00e4v\u00e4 216. a)<\/div>\n\n\n\n
Kuutio tarkoittaa kolmatta potenssia ja neli\u00f6 tarkoittaa toista potenssia. Siten k\u00e4yr\u00e4n yht\u00e4l\u00f6 on muotoa<\/p>\n\n\n\n
<\/span> <\/span>![\"\[y^3=x^2\]\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-f9de4acd8e324150aadc9c5c0ea83f7b_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nPiirret\u00e4\u00e4n k\u00e4yr\u00e4 GeoGebralla.<\/p>\n\n\n\n
![\"\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/uploads\/sites\/136\/2019\/08\/216_1.png\")
K\u00e4yr\u00e4 ![\"y^3=x^2\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-aaaeb7bf6c353a414a04f74cd0e68391_l3.png\" "\\"Rendered")
<\/figcaption><\/figure><\/div>\n\n\n\nTeht\u00e4v\u00e4 216. b)<\/div>\n\n\n\n
Lasketaan :n arvot, kun ![\"y=2\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-701efe280cdb8ebc9f2d73ddf9b76c07_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\n
<\/span> <\/span>![\"\begin{align*}2^3&=x^2\\x^2&=8\ \parallel\ \sqrt{\ }\\x&=\sqrt{8}=\sqrt{4 \cdot 2}=\sqrt{4} \cdot \sqrt{2}=2\sqrt{2}\\\text{tai}\\ x&=-\sqrt{8}=-\sqrt{4 \cdot 2}=-\sqrt{4} \cdot \sqrt{2}=-2\sqrt{2}\end{align*}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-7bb5beebe01e293114320fc668eae22c_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nT\u00e4m\u00e4n voisi kirjoittaa my\u00f6s lyhyemmin seuraavasti:<\/p>\n\n\n\n
<\/span> <\/span>![\"\begin{align*}2^3&=x^2\\x^2&=8\ \parallel\ \sqrt{\ }\\x&=\pm \sqrt{8}=\pm \sqrt{4 \cdot 2}=\pm \sqrt{4} \cdot \sqrt{2}=\pm 2\sqrt{2}\\\end{align*}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-10a0af824f1f6cc6da87ec1bb91d2701_l3.png\" "\\"Rendered")
<\/p> <\/p>\n\n\n\nKysytyt pisteet ovat ![\"\left( 2\sqrt{2}, 2\right)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-be2d4cea904360c3689d7075d634c8c5_l3.png\" "\\"Rendered")
ja ![\"\left( -2\sqrt{2}, 2\right)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-4e7c92cb3fdc3e575d4ed5a1bda2815e_l3.png\" "\\"Rendered")
.<\/p>\n\n\n\n
Teht\u00e4v\u00e4 216. c)<\/div>\n\n\n\n
Jos ![\"y < 0\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-45b8931e701cc17ba3504c982e846f06_l3.png\" "\\"Rendered")
, niin ![\"y^3 < 0\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-dff7be130647172b04e1a69237667606_l3.png\" "\\"Rendered")
. T\u00e4ll\u00f6in olisi l\u00f6ydett\u00e4v\u00e4 sellainen -koordinaatti,
ett\u00e4 sen neli\u00f6 on negatiivinen. T\u00e4llaisia ei ole, joten yht\u00e4l\u00f6n ![\"y^3 = x^2\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-13d6a42c3a5a83f9b4edee532c44b777_l3.png\" "\\"Rendered")
toteuttavien pisteiden joukossa yhdenk\u00e4\u00e4n pisteen -koordinaatti ei ole
negatiivinen.
<\/p>\n","protected":false},"excerpt":{"rendered":"
T\u00e4ll\u00e4 sivulla on Tuntimuistiinpanot luvusta 2.1 tiistaina 24.8.2021 Luku 2.1: Teht\u00e4vien 204, 210, 213, 214, 215, 216 ratkaisut. Mukana on my\u00f6s teht\u00e4v\u00e4 210, vaikka teimme sen jo tunneilla. Teht\u00e4v\u00e4 204. A – II, B – III, C – IV ja D – I Teht\u00e4v\u00e4 210. a) Pisteen et\u00e4isyys pisteest\u00e4 : Pisteen et\u00e4isyys pisteest\u00e4 : Pisteen … Jatka lukemista 2.1 (ti 24.8.)<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":111,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_uag_custom_page_level_css":"","footnotes":""},"class_list":["post-976","page","type-page","status-publish","hentry"],"yoast_head":"\n2.1 (ti 24.8.) - Analyyttinen geometria (MAA05)<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogit.gradia.fi\/ageom\/25-8-ti\/\" \/>\n<meta property=\"og:locale\" content=\"fi_FI\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"2.1 (ti 24.8.) - Analyyttinen geometria (MAA05)\" \/>\n<meta property=\"og:description\" content=\"T\u00e4ll\u00e4 sivulla on Tuntimuistiinpanot luvusta 2.1 tiistaina 24.8.2021 Luku 2.1: Teht\u00e4vien 204, 210, 213, 214, 215, 216 ratkaisut. Mukana on my\u00f6s teht\u00e4v\u00e4 210, vaikka teimme sen jo tunneilla. Teht\u00e4v\u00e4 204. 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![\"B = (1, 2)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-a3697ffc53a9b8ab89e000f16e9f498e_l3.png\" "\\"Rendered")
et\u00e4isyys pisteest\u00e4 ![\"A = (2, 1)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-84330613d1e13a8dfa997007788cf2ed_l3.png\" "\\"Rendered")
:![\"\sqrt{(2-1)^2+(1-2)^2}=\sqrt{1^2+(-1)^2}=\sqrt{1+1}=\sqrt{2}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-0cee26c8369a972782e2452be7e9ab02_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"C = (3, 3)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-08cc57f65923ce6e0595e8ea560fe6f6_l3.png\" "\\"Rendered")
et\u00e4isyys pisteest\u00e4 ![\"\sqrt{(2-3)^2+(1-3)^2}=\sqrt{(-1)^2+(-2)^2}=\sqrt{1+4}=\sqrt{5}\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-286163262c0135288f6ab6ae4a27d6cc_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"D = (5, 1)\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-9a7080b349daf2cb0e6534cbc21572db_l3.png\" "\\"Rendered")
et\u00e4isyys pisteest\u00e4 ![\"\sqrt{(2-5)^2+(1-1)^2}=\sqrt{(-3)^2+0^2}=\sqrt{9+0}=\sqrt{9}=3\"](\"https:\/\/blogit.gradia.fi\/ageom\/wp-content\/ql-cache\/quicklatex.com-d97ab20332548cb261302349c25db6fb_l3.png\" "\\"Rendered")
<\/p>\n\n\n\nTeht\u00e4v\u00e4 210. b)<\/div>\n\n\n\n