現在地と前後の項目 *** 基本 ***/三平方の定理1/三平方の定理2/三平方の定理(徹底復習)/*** 距離 ***/2点間の距離1/2点間の距離2/2点間の距離3/震源地/*** 逆の問題 ***/三平方の定理の逆/*** 融合問題 ***/融合問題/問題以上,答以下/展開図形/回転図形1/回転図形2/円と三平方の定理/*** よく出る応用 ***/三角形2個の問題1/三角形2個の問題2/三角形2個の問題3/特別な形の三角形1/特別な形の三角形2/三辺→高さ/*** 空間図形 ***/空間図形と三平方の定理1/空間図形と三平方の定理2/立体の体積,表面積/立体の体積(入試問題)/立体の表面積.展開図(入試問題)/
【三平方の定理】
○三平方の定理を使えば,直角三角形の2辺の長さが分かれば残りの1辺の長さが求められる.右図のような直角三角形については
b2+c2=a2
が成り立つ.
たとえば右図では,
b , c が分かっていれば a が求められる.
a , c が分かっていれば b が求められる. a , b が分かっていれば c が求められる. |
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例1 右図で黄色の三角形について三平方の定理を使うとAHの長さが求めることができ,
さらに求めたAHの長さとCHの長さを使って青色の三角形について三平方の定理を使うと辺ACの長さを求めることができる.
22+x2=(2
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4+x2=20
x2=16
x=4 (>0)
さらに 42+32=y2
y2=25
y=5 (>0)
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問題1 右図において x , y の長さを求めなさい.
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![]() ( ![]() ![]()
7+x2=32
x2=25
x=5 (>0) ( ![]()
y2=36
y=6 (>0) |
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問題2 右図において CD の長さ x を求めなさい.
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![]() 12+AB2=( ![]()
AB2=4
AB=2 (>0)
直角三角形ABD について三平方の定理を適用すると
22+(x+1)2=( ![]()
(x+1)2=9
x+1=3 (>0) x=2 |
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例2 長方形の向かい合う辺の長さは等しいので,右図で AH=DC になる.
このAH の長さとAB の長さから三角形ABH について三平方の定理を使うと辺BH が求まり,HC,AD の長さが分かる.
AH=DC=
△ABH に三平方の定理を適用すると
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(
![]() ![]()
3+BH2=7
BH2=4
したがって
BH=2 (>0)
CH=3
x=3
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問題3 右図において CD の長さ x を求めなさい.
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HC=AD=5 , BH=2
直角三角形ABH に三平方の定理を適用すると
BH2+AH2=AB2
22+x2=(2 ![]() 4+x2=20 x2=16 x=4 (>0) |
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直角三角形ABC に三平方の定理を適用すると
202+152=AC2
△ABC と△ADB は直角三角形で1つの角A が共通だから「2つの角がそれぞれ等しい」.したがって△ABC∽△ADBAC2=400+225=625=252 AC=25 (>0) 相似三角形の2組の辺の比は等しいから
AC:CB=AB:x
25:20=15:x 25x=300 x=12 (別解) この問題については,△ABC の面積S を2通りの方法で表して,それらが等しいことからx を求める方法もある.(1つの方法で行き詰ったときのために,別ルートも覚えておくと助かることがある.)
AB · BC / 2 = S = AC · x / 2
AB · BC = 2S = AC · x 15 · 20 = 2S = 25 · x x=12 |
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問題4 右図において BD の長さ x を求めなさい.
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AB2+BC2=AC2
△ACB∽△ABD だから
( ![]() 5+BC2=25 BC2=20 BC=2 ![]()
AC:CB=AB:x
5:2 ![]() ![]() 5x=10 x=2 |
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例4 二等辺三角形では,頂点から底辺に引いた垂線は底辺を二等分するので,直角三角形を使って高さを求めることができる.
△ABC はAB=AC の二等辺三角形だから,A からBC に垂線BE をひくとAE⊥BC,BE=EC=3 になる.さらに相似図形の性質を使えば,右図 DC の長さも求められる. 直角三角形ABE に三平方の定理を適用すると
32+AE2=92
△ABE と△BCD は直角三角形で1つの角B が共通だから「2つの角がそれぞれ等しい」.したがって△ABE∽△BCDAE2=72 AE=6 ![]() 相似三角形の2組の辺の比は等しいから
BA:AE=BC:CD
9:6 ![]() 9x=36 ![]() x=4 ![]() (別解) △ABC の面積をS とおくと
AE · BC / 2 = S = AB · x / 2
6 ![]() 両辺を2倍して分母をはらうと 36 ![]() x=4 ![]() |
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問題5 右図の二等辺三角形ABC において頂点 C からAB にひいた垂線の長さ x を求めなさい.
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![]() ![]() △ABE に三平方の定理を適用すると
(
△ABE∽△BCD だから
![]() 3+AE2=9 AE2=6 AE= ![]()
BA:AE=BC:x
3: ![]() ![]() 3x= ![]() ![]() ![]() x=2 ![]() |
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