Tambayoyi Misali Game da Sunaye Gefen Alwatika Na Dama
Pendahuluan
Alwatika mai kyau alwatika ce mai kusurwar digiri 90. Wannan alwatika tana da matuƙar muhimmanci a fannin lissafi da aikace-aikacenta daban-daban, ciki har da kimiyyar lissafi, injiniyancin farar hula, da sauran fannoni daban-daban na kimiyya. Ɗaya daga cikin muhimman abubuwan da ke tattare da nazarin alwatika masu kyau shine fahimtar sunayen kowane gefe da kuma yadda ake gane su. Wannan labarin zai yi bayani dalla-dalla kan misalan matsaloli kuma ya tattauna yadda ake sanya sunayen ɓangarorin alwatika mai kyau dalla-dalla.
Sunaye Gefen Alwatika Na Dama
A cikin alwatika na dama, akwai ɓangarori uku waɗanda ke da sunaye na musamman:
1. Hauhawar jini: Wannan shine mafi tsayin gefen a cikin alwatika na dama kuma koyaushe yana adawa da kusurwar dama.
2. Tushe: Ɗaya daga cikin ɓangarorin biyu da suka samar da kusurwar dama.
3. Gefen Perpendicular (Tsawo/Daidaitacce): Ɗaya daga cikin ɓangarorin biyu da ke samar da kusurwar dama kuma yawanci ana ɗaukarsa a matsayin daidaitacce ga tushe.
Misali Tambaya ta 1: Gano Gefen Alwatika Mai Daidai
Tambaya:
An ba da alwatika ABC mai kusurwar dama a B. Tsawon AB shine 3 cm, tsawon BC shine 4 cm, kuma tsawon AC shine 5 cm. Ka tantance sunan kowane gefen alwatika.
Tattaunawa:
1. Ƙayyade Hauhawar Jini:
Haushin haila shine mafi tsayin gefe a cikin alwatika na dama, wanda ke gaban kusurwar dama (∠B). Tsawon AC = 5 cm shine mafi tsayin gefe, don haka AC shine haushin haila.
2. Kayyade gefen tushe da gefen tsaye:
Gefen biyu da suka samar da kusurwar dama sune AB da BC. Ta hanyar kwatanta tsayinsu, BC (4 cm) da AB (3 cm), za mu iya cewa AB, wanda ya fi guntu, shine gefen da ke tsaye, BC kuma shine tushe.
Don haka, sakamakon sanya wa bangarorin suna shine:
– Hypotenuse: AC
– Gefen Tushe: BC
– Gefen Tsaye: AB
Misali Tambaya ta 2: Lissafin Tsawon Gefen Alwatika na Dama Ta Amfani da Ka'idar Pythagorean
Tambaya:
An ba da alwatika mai kusurwar dama a E. Tsawon DE shine 6 cm kuma tsawon EF shine 8 cm. Lissafa tsawon gefen DF (hypotenuse).
Tattaunawa:
Don ƙididdige tsawon hypotenuse (DF), za mu iya amfani da Ka'idar Pythagorean, wadda ta bayyana cewa a cikin alwatika mai kusurwa huɗu:
\[ \text{Hypotenuse}^2 = \text{Base Gefen}^2 + \text{Pendendicular Gefen}^2 \]
A cikin wannan tambayar:
– DE da EF su ne ɓangarorin da ke samar da kusurwar dama, don haka DE da EF su ne ɓangarorin tushe da kuma a tsaye.
– DE = 6 cm da EF = 8 cm.
Amfani da ka'idar Pythagorean:
\[ DF^2 = DE^2 + EF^2 \]
\[ DF^2 = 6^2 + 8^2 \]
\[ DF^2 = 36 + 64 \]
\[ DF^2 = 100 \]
Shan tushen murabba'i na ɓangarorin biyu:
\[ DF = \sqrt{100} \]
\[ DF = 10 \rubutu{ cm} \]
Don haka, tsawon hypotenuse DF shine 10 cm.
Misali na 3: Tantance Tsawon Gefen Daidaitacce Ta Amfani da Ka'idar Pythagorean
Tambaya:
Alwatika MNO alwatika ce ta dama mai kusurwar dama a N. Tsawon MN shine 9 cm kuma tsawon hypotenuse MO shine 15 cm. Lissafa tsawon NO na gefe.
Tattaunawa:
Daga tambayar, mun san cewa:
– MN yana ɗaya daga cikin ɓangarorin da ke samar da kusurwar dama (gefen tsaye).
– MO shine hypotenuse.
Ta amfani da Ka'idar Pythagorean don nemo tsawon NO:
\[ \text{Hypotenuse}^2 = \text{Base Gefen}^2 + \text{Pendendicular Gefen}^2 \]
\[ 15^2 = 9^2 + NO^2 \]
\[ 225 = 81 + NO^2 \]
Ware NO^2:
\[ NO^2 = 225 – 81 \]
\[ NO^2 = 144 \]
Shan tushen murabba'i na ɓangarorin biyu don samun NO:
\[ A'a = \sqrt{144} \]
\[ NO = 12 \rubutu{ cm} \]
Don haka, tsawon NO na gefen shine 12 cm.
Misali Tambaya ta 4: Tantance Gefen Tushe Ta Amfani da Ka'idar Pythagorean
Tambaya:
PQR mai kusurwa uku mai kusurwar dama yana da tsawon PR (hypotenuse) na 13 cm da tsawon PQ (gefen gefe mai kusurwa) na 5 cm. Lissafa tsawon QR na gefe (gefen tushe).
Tattaunawa:
Amfani da Ka'idar Pythagorean:
\[ \text{Hypotenuse}^2 = \text{Base Gefen}^2 + \text{Pendendicular Gefen}^2 \]
\[ 13^2 = QR^2 + 5^2 \]
\[ 169 = QR^2 + 25 \]
Ware QR^2:
\[ QR^2 = 169 – 25 \]
\[ QR^2 = 144 \]
Ɗauki tushen murabba'in ɓangarorin biyu don nemo QR:
\[ QR = \sqrt{144} \]
\[ QR = 12 \rubutu{ cm} \]
Don haka, tsawon QR na gefe shine 12 cm.
Kammalawa
Ta hanyar nazarin misalan da ke sama, za mu iya gano da kuma fahimtar sunayen bangarorin alwatika na dama, da kuma amfani da Ka'idar Pythagorean don ƙididdige tsawon gefen da ba a sani ba. Wannan ilimin yana da mahimmanci don magance matsalolin lissafi masu rikitarwa da aikace-aikacensa a fannoni daban-daban. Fahimtar waɗannan mahimman ra'ayoyi zai ba wa ɗalibai damar magance ƙalubalen da suka shafi yanayin alwatika cikin inganci.
