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2018 AMC 10A Problems/Problem 13: Difference between revisions

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A paper triangle with sides of lengths 3,4, and 5 inches, as shon, is folded so that point <math>A</math> falls on point <math>B</math>. What is the length in inches of the crease?
A paper triangle with sides of lengths 3,4, and 5 inches, as shon, is folded so that point <math>A</math> falls on point <math>B</math>. What is the length in inches of the crease?
[asy]
<asy>
draw((0,0)--(4,0)--(4,3)--(0,0));
draw((0,0)--(4,0)--(4,3)--(0,0));
label("<math>A</math>", (0,0), SW);
label("$A$", (0,0), SW);
label("<math>B</math>", (4,3), NE);
label("$B$", (4,3), NE);
label("<math>C</math>", (4,0), SE);
label("$C$", (4,0), SE);
label("<math>4</math>", (2,0), S);
label("$4$", (2,0), S);
label("<math>3</math>", (4,1.5), E);
label("$3$", (4,1.5), E);
label("<math>5</math>", (2,1.5), NW);
label("$5$", (2,1.5), NW);
fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray);
fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray);
[/asy]
</asy>
<math>\textbf{(A) }  1+\frac12 \sqrt2  \qquad        \textbf{(B) }  \sqrt3  \qquad    \textbf{(C) }  \frac74  \qquad  \textbf{(D) }  \frac{15}{8} \qquad  \textbf{(E) }  2 </math>
<math>\textbf{(A) }  1+\frac12 \sqrt2  \qquad        \textbf{(B) }  \sqrt3  \qquad    \textbf{(C) }  \frac74  \qquad  \textbf{(D) }  \frac{15}{8} \qquad  \textbf{(E) }  2 </math>

Revision as of 13:03, 8 February 2018

A paper triangle with sides of lengths 3,4, and 5 inches, as shon, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? [asy] draw((0,0)--(4,0)--(4,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,3), NE); label("$C$", (4,0), SE); label("$4$", (2,0), S); label("$3$", (4,1.5), E); label("$5$", (2,1.5), NW); fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray); [/asy] $\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2$

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