*** 蟄ヲ蟷エ *** 荳ュ蟄ヲ�大ケエ荳ュ蟄ヲ�貞ケエ荳ュ蟄ヲ�灘ケエ *** 蜊伜� *** 蠑上�螻暮幕蝗�謨ー蛻�ァ」邏�謨ー繝サ邏�蝗�謨ー蛻�ァ」蟷ウ譁ケ譬ケ �呈ャ。譁ケ遞句シ�荳牙ケウ譁ケ縺ョ螳夂炊�呈ャ。髢「謨ー蟷ウ陦檎キ壹�逶ク莨シ 窶サ荳ュ蟄ヲ�灘ケエ逕溷髄縺�縲御ク牙ケウ譁ケ縺ョ螳夂炊縲�縺ォ縺、縺�※�後%縺ョ繧オ繧、繝医↓縺ッ谺。縺ョ謨呎攝縺後≠繧翫∪縺呻シ� 縺薙�鬆√∈Google繧ШAHOO ! 縺ェ縺ゥ縺ョ讀懃エ「縺九i逶エ謗・譚・縺ヲ縺励∪縺」縺溘�縺ァ縲悟燕謠舌→縺ェ縺」縺ヲ縺�k蜀�ョケ縺悟�縺九i縺ェ縺�阪→縺�≧蝣エ蜷医d縲後%縺ョ鬆√�蛻�°縺」縺溘′繧ゅ▲縺ィ蠢懃畑蝠城。後r隕九◆縺�阪→縺�≧蝣エ蜷医��御サ悶�鬆√r隕九※縺上□縺輔>��縲 縺檎樟蝨ィ蝨ー縺ァ縺呻シ� ![]() ![]() ■次のような直角三角形の三辺の長さについては, ![]() (これを三平方の定理の逆といいます.) 一番長い辺が斜辺です. ※ 直角三角形であるかどうかを調べるには,
【要点】
小さい方の2辺を直角な2辺として,2乗の和 a2+b2 を作り,一番長い辺を斜辺として c2 を作る.
a2+b2=c2 ⇒ 直角三角形
a2+b2≠c2 ⇒ 直角三角形ではない |
■問題1 次のように三角形の三辺の長さが与えられているとき,これらのうちで直角三角形となっているものを選びなさい.
(4組のうち1組が直角三角形です.)
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(4)
「6 , 11 , 13」 「6 , 12 , 13」 |
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(5)
「9 , 39 , 41」 「9 , 40 , 41」 |
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■問題2 次のように三角形の三辺の長さが与えられているとき,これらのうちで直角三角形となっているものを選びなさい. |
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( ![]() ![]() ![]() ![]() ![]() ![]() =42 |
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(3
![]() ![]() ![]() ![]() =(2 ![]() ( ![]() ![]() ![]() ![]() ![]() ![]() =(3 ![]() |
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(2
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左辺は 4m2n2+m4-2m2n2+n4 右辺は m4+2m2n2+n4 だから等しい 例 m=2,n=1 を代入すると 42+32=52 となります.(このとき,3 , 4 , 5 の組がピタゴラス数) |
■(自由研究) 左の式を利用して,三辺の長さが整数となる直角三角形を1組見つけなさい.(上の問題にないもので答えなさい・・・ただし,このホームページでは,あまり大きな数字の計算はできないので,どの辺の長さも100以下で答えなさい.) |
ピタゴラス数の例(小さい方から幾つか) |
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m | n | 2mn | m2−n2 | m2+n2 | 成り立つ関係 |
2 | 1 | 4 | 3 | 5 | 42+32=52 |
3 | 1 | 6 | 8 | 10 | 62+82=102 |
3 | 2 | 12 | 5 | 13 | 122+52=132 |
4 | 1 | 8 | 15 | 17 | 82+152=172 |
4 | 2 | 16 | 12 | 20 | 162+122=202 |
4 | 3 | 24 | 7 | 25 | 242+72=252 |
5 | 1 | 10 | 24 | 26 | 102+242=262 |
5 | 2 | 20 | 21 | 29 | 202+212=292 |
5 | 3 | 30 | 16 | 34 | 302+162=342 |
5 | 4 | 40 | 9 | 41 | 402+92=412 |
6 | 1 | 12 | 35 | 37 | 122+352=372 |
6 | 2 | 24 | 32 | 40 | 242+322=402 |
6 | 3 | 36 | 27 | 45 | 362+272=452 |
6 | 4 | 48 | 20 | 52 | 482+202=522 |
6 | 5 | 60 | 11 | 61 | 602+112=612 |
7 | 1 | 14 | 48 | 50 | 142+482=502 |
7 | 2 | 28 | 45 | 53 | 282+452=532 |
7 | 3 | 42 | 40 | 58 | 422+402=582 |
7 | 4 | 56 | 33 | 65 | 562+332=652 |
7 | 5 | 70 | 24 | 74 | 702+242=742 |
7 | 6 | 84 | 13 | 85 | 842+132=852 |
8 | 1 | 16 | 63 | 65 | 162+632=652 |
8 | 2 | 32 | 60 | 68 | 322+602=682 |
8 | 3 | 48 | 55 | 73 | 482+552=732 |
8 | 4 | 64 | 48 | 80 | 642+482=802 |
8 | 5 | 80 | 39 | 89 | 802+392=892 |
8 | 6 | 96 | 28 | 100 | 962+282=1002 |
8 | 7 | 112 | 15 | 113 | 1122+152=1132 |
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