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Difference between revisions of "1996 AJHSME Problems/Problem 22"
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==Problem==
 
==Problem==
  
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The horizontal and vertical distances between adjacent points equal 1 unit.  The area of triangle <math>ABC</math> is
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The horizontal and vertical distances between adjacent points equal 1 unit.  What is the area of triangle <math>ABC</math>?
  
 
<asy>
 
<asy>
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}
 
}
 
draw((0,0)--(3,2)--(4,3)--cycle);
 
draw((0,0)--(3,2)--(4,3)--cycle);
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label("$A$",(0,0),SW);wwqaerhqe
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label("$A$",(0,0),SW);
 
label("$B$",(3,2),SE);
 
label("$B$",(3,2),SE);
 
label("$C$",(4,3),NE);
 
label("$C$",(4,3),NE);
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Thus, the area of <math>\triangle ABC = 6 - 4 - \frac{3}{2} = \frac{1}{2}</math>, and the answer is <math>\boxed{B}</math>.
 
Thus, the area of <math>\triangle ABC = 6 - 4 - \frac{3}{2} = \frac{1}{2}</math>, and the answer is <math>\boxed{B}</math>.
  
−
There are other equivalent ways of dissecting the figure; right triangles <math>\triangle ABF, \triangle BCE</math> and rectangle <math>\square BEDF</math> can also be used.
+
There are other equivalent ways of dissecting the figure; right triangles <math>\triangle ABF, \triangle BCE</math> and rectangle <math>\square BEDF</math> can also be used. You can also use <math>\triangle{BEC}</math> and trapezoid <math>ADBE</math>.
  
 
==Solution 2==
 
==Solution 2==
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Area = <math>\frac{1}{2} = |(3\cdot 0 + 4\cdot 2 + 0\cdot 3) - (0\cdot 2 + 3\cdot 3 + 4\cdot 0)| = \frac{1}{2}</math>, which is option <math>\boxed{B}</math>.
 
Area = <math>\frac{1}{2} = |(3\cdot 0 + 4\cdot 2 + 0\cdot 3) - (0\cdot 2 + 3\cdot 3 + 4\cdot 0)| = \frac{1}{2}</math>, which is option <math>\boxed{B}</math>.
 +
 +
==Solution 3==
 +
Using [[Pick's Theorem]], we can simply use the number of coordinate points to get the formula. In application, we get the formula <math>\frac{3}{2} + 0 - 1</math>, which equals <math>\frac{1}{2}</math>, giving us our answer <math>\boxed{B}</math>.
  
 
==See Also==
 
==See Also==
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* [[AJHSME Problems and Solutions]]
 
* [[AJHSME Problems and Solutions]]
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
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{{MAA Notice}}

Latest revision as of 10:31, 27 June 2023

Problem

The horizontal and vertical distances between adjacent points equal 1 unit. What is the area of triangle $ABC$?

[asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 4; ++b) { dot((a,b)); } } draw((0,0)--(3,2)--(4,3)--cycle); label("$A$",(0,0),SW); label("$B$",(3,2),SE); label("$C$",(4,3),NE); [/asy]

$\text{(A)}\ 1/4 \qquad \text{(B)}\ 1/2 \qquad \text{(C)}\ 3/4 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 5/4$

Solution 1

[asy] for (int a = 0; a < 5; ++a) { for (int b = 0; b < 4; ++b) { dot((a,b)); } } draw((0,0)--(3,2)--(4,3)--cycle); draw((0,0)--(3,2)--(4,0)--cycle); draw((4,2)--(3,2)--(4,3)--cycle); draw((0,0)--(4,0)--(4,3)--cycle); draw((3,2)--(3,0)--cycle); label("$A$",(0,0),SW); label("$B$",(3,2),SE); label("$C$",(4,3),NE); label("$D$",(4,0),SE); label("$E$",(4,2),SE); label("$F$",(3,0),SE); [/asy]

$\triangle ADC$ takes up half of the 4x3 grid, so it has area of $6$.

$\triangle ABD$ has height of $BF = 2$ and a base of $AD = 4$, for an area of $\frac{1}{2}\cdot 2 \cdot 4 = 4$.

$\triangle CBD$ has height of $BE = 1$ and a base of $CD = 3$, for an area of $\frac{1}{2}\cdot 1 \cdot 3 = \frac{3}{2}$

Note that $\triangle ABC$ can be found by taking $\triangle ADC$, and subtracting off $\triangle ABD$ and $\triangle CBD$.

Thus, the area of $\triangle ABC = 6 - 4 - \frac{3}{2} = \frac{1}{2}$, and the answer is $\boxed{B}$.

There are other equivalent ways of dissecting the figure; right triangles $\triangle ABF, \triangle BCE$ and rectangle $\square BEDF$ can also be used. You can also use $\triangle{BEC}$ and trapezoid $ADBE$.

Solution 2

Using the Shoelace Theorem, and labelling the points $A(0,0), B(3,2), C(4,3)$, we find the area is:

$(0,0)$

$(3,2)$

$(4,3)$

$(0,0)$

Area = $\frac{1}{2} = |(3\cdot 0 + 4\cdot 2 + 0\cdot 3) - (0\cdot 2 + 3\cdot 3 + 4\cdot 0)| = \frac{1}{2}$, which is option $\boxed{B}$.

Solution 3

Using Pick's Theorem, we can simply use the number of coordinate points to get the formula. In application, we get the formula $\frac{3}{2} + 0 - 1$, which equals $\frac{1}{2}$, giving us our answer $\boxed{B}$.

See Also

1996 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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