已知等腰三角形的兩條邊分別為5,6,求一腰上的高線長.
解:
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△ABC中,AB=AC,
設(shè)AD=x,
分為兩種情況:①當(dāng)AB=AC=5,BC=6時,
則CD=5-x,
∵在Rt△ABD和Rt△CDB中,由勾股定理得:BD
2=AB
2-AD
2=BC
2-CD
2,
∴5
2-x
2=6
2-(5-x)
2,
x=
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,
∴BD
2=5
2-(
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)
2,
∴BD=
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,
②當(dāng)AB=AC=6,BC=5時,
則CD=6-x,
∵在Rt△ABD和Rt△CDB中,由勾股定理得:BD
2=AB
2-AD
2=BC
2-CD
2,
∴6
2-x
2=5
2-(6-x)
2,
x=
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,
∴BD
2=6
2-(
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)
2,
∴BD=
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;
即一腰上的高線長是
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或
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.
分析:設(shè)AD=x,分為兩種情況:①當(dāng)AB=AC=5,BC=6時,②當(dāng)AB=AC=6,BC=5時,由勾股定理得:BD
2=AB
2-AD
2=BC
2-CD
2,代入求出x,把x的值代入BD
2=AB
2-AD
2求出即可.
點評:本題考查了等腰三角形的性質(zhì),勾股定理等知識點,主要考查學(xué)生的計算能力和推理能力,注意有兩個解.