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Difference between revisions of "1999 AHSME Problems/Problem 2"

(New page: By definition of equilateral triangles, only <math>\boxed{\text{A}}</math> is false.)
 
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By definition of equilateral triangles, only <math>\boxed{\text{A}}</math> is false.
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==Problem==
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Which of the following statements is false?
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<math> \mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}</math>
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<math>\mathrm{(B) \  All\ equilateral\ triangles\ are\ convex.}</math>
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<math>\mathrm{(C) \  All\ equilateral\ triangles\ are\ equianguilar.}</math>
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<math>\mathrm{(D) \  All\ equilateral\ triangles\ are\ regular\ polygons.}</math>
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<math>\mathrm{(E) \  All\ equilateral\ triangles\ are\ similar\ to\ each\ other.}  </math>
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==Solutions==
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===Solution 1===
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An equilateral triangle is isosceles, and we find that <math>\angle A=\angle B=\angle C</math> if we use the property of isosceles triangles that if two sides of a triangle are equal then the opposite angles are equal. Thus equilateral triangles are equiangular. Regular pentagons are both equilateral and equiangular, and so are equilateral triangles. Thus equilateral triangles are regular polygons. Since all of the angles are the same, all equilateral triangles are similar. Since <math>\angle A=\angle B=\angle C</math> and <math>\angle A+\angle B+\angle C=180</math>, <math>\angle A=60^{\circ}</math>. Since no other angles are above <math>180^{\circ}</math>, all equilateral triangles are convex. This just leaves choice <math>\boxed{\mathrm{(A)}}</math>.
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===Solution 2===
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Congruent triangles have the same side length. <geogebra>f1a4134248dedfd1e182804e035fd400ef763f04</geogebra>  The image above disproves <math>\boxed{\mathrm{(A)}}</math>.
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==See also==
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[[Category:Introductory Geometry Problems]]

Revision as of 13:32, 1 December 2008

Problem

Which of the following statements is false?

$\mathrm{(A) \ All\ equilateral\ triangles\ are\ congruent\ to\ each\ other.}$ $\mathrm{(B) \ All\ equilateral\ triangles\ are\ convex.}$ $\mathrm{(C) \ All\ equilateral\ triangles\ are\ equianguilar.}$ $\mathrm{(D) \ All\ equilateral\ triangles\ are\ regular\ polygons.}$ $\mathrm{(E) \ All\ equilateral\ triangles\ are\ similar\ to\ each\ other.}$

Solutions

Solution 1

An equilateral triangle is isosceles, and we find that $\angle A=\angle B=\angle C$ if we use the property of isosceles triangles that if two sides of a triangle are equal then the opposite angles are equal. Thus equilateral triangles are equiangular. Regular pentagons are both equilateral and equiangular, and so are equilateral triangles. Thus equilateral triangles are regular polygons. Since all of the angles are the same, all equilateral triangles are similar. Since $\angle A=\angle B=\angle C$ and $\angle A+\angle B+\angle C=180$, $\angle A=60^{\circ}$. Since no other angles are above $180^{\circ}$, all equilateral triangles are convex. This just leaves choice $\boxed{\mathrm{(A)}}$.

Solution 2

Congruent triangles have the same side length. <geogebra>f1a4134248dedfd1e182804e035fd400ef763f04</geogebra> The image above disproves $\boxed{\mathrm{(A)}}$.

See also

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