https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Andrewaudrey02&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T19:06:07ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_21&diff=1357782019 AMC 8 Problems/Problem 212020-10-25T00:36:54Z<p>Andrewaudrey02: /* Solution 2 */</p>
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==Problem 21==<br />
What is the area of the triangle formed by the lines <math>y=5</math>, <math>y=1+x</math>, and <math>y=1-x</math>?<br />
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<math>\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16</math><br />
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==Solution 1==<br />
First we need to find the coordinates where the graphs intersect. <br />
<br />
<math>y=5</math>, and <math>y=x+1</math> intersect at <math>(4,5)</math>, <br />
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<math>y=5</math>, and <math>y=1-x</math> intersect at <math>(-4,5)</math>,<br />
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<math>y=1-x</math> and <math>y=1+x</math> intersect at <math>(0,1)</math>. <br />
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Using the [[Shoelace Theorem]] we get: <cmath>\left(\frac{(20-4)-(-20+4)}{2}\right)=\frac{32}{2}</cmath> <math>=</math> So our answer is <math>\boxed{\textbf{(E)}\ 16}</math>.<br />
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==Solution 2==<br />
Graphing the lines, using the intersection points we found in Solution 1, we can see that the height of the triangle is 4, and the base is 8. Using the formula for the area of a triangle, we get <math>\frac{4\cdot8}{2}</math> which is equal to <math>\boxed{\textbf{(E)}\ 16}</math>.<br />
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==Video Solutions==<br />
https://www.youtube.com/watch?v=9nlX9VCisQc<br />
<br />
https://www.youtube.com/watch?v=mz3DY1rc5ao<br />
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https://www.youtube.com/watch?v=Z27G0xy5AgA&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=3 ~ MathEx<br />
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https://www.youtube.com/watch?v=aStuVhoD8wc- Also includes other problems from 21-25<br />
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==See Also==<br />
{{AMC8 box|year=2019|num-b=20|num-a=22}}<br />
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{{MAA Notice}}</div>Andrewaudrey02https://artofproblemsolving.com/wiki/index.php?title=2019_AMC_8_Problems/Problem_21&diff=1357772019 AMC 8 Problems/Problem 212020-10-25T00:36:26Z<p>Andrewaudrey02: /* Solution 1 */</p>
<hr />
<div><br />
==Problem 21==<br />
What is the area of the triangle formed by the lines <math>y=5</math>, <math>y=1+x</math>, and <math>y=1-x</math>?<br />
<br />
<math>\textbf{(A) }4\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }16</math><br />
<br />
==Solution 1==<br />
First we need to find the coordinates where the graphs intersect. <br />
<br />
<math>y=5</math>, and <math>y=x+1</math> intersect at <math>(4,5)</math>, <br />
<br />
<math>y=5</math>, and <math>y=1-x</math> intersect at <math>(-4,5)</math>,<br />
<br />
<math>y=1-x</math> and <math>y=1+x</math> intersect at <math>(0,1)</math>. <br />
<br />
Using the [[Shoelace Theorem]] we get: <cmath>\left(\frac{(20-4)-(-20+4)}{2}\right)=\frac{32}{2}</cmath> <math>=</math> So our answer is <math>\boxed{\textbf{(E)}\ 16}</math>.<br />
<br />
==Solution 2==<br />
Graphing the lines, using the intersection points we found in Solution 1, we can see that the height of the triangle is 4, and the base is 8. Using the formula for the area of a triangle, we get <math>\frac{4\cdot8}{2}</math> which is equal to <math>\boxed{\textbf{(E)}\ 16}</math>. <br />
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~SmileKat32<br />
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~more edits by BakedPotato69<br />
<br />
==Video Solutions==<br />
https://www.youtube.com/watch?v=9nlX9VCisQc<br />
<br />
https://www.youtube.com/watch?v=mz3DY1rc5ao<br />
<br />
https://www.youtube.com/watch?v=Z27G0xy5AgA&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=3 ~ MathEx<br />
<br />
https://www.youtube.com/watch?v=aStuVhoD8wc- Also includes other problems from 21-25<br />
<br />
==See Also==<br />
{{AMC8 box|year=2019|num-b=20|num-a=22}}<br />
<br />
{{MAA Notice}}</div>Andrewaudrey02