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Difference between revisions of "2024 AMC 8 Problems/Problem 24"

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m (→‎Solution 1)
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==Solution 1==
 
==Solution 1==
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Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle. The side length of the largest right triangle is <math>12\sqrt{2},</math> which means its area is <math>144.</math> Similarly, the area of the second largest right triangle is <math>64,</math> and the area of the overlap triangle is <math>h^2.</math> Thus,
+
Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle. The side length of the largest right triangle is <math>12\sqrt{2},</math> which means its area is <math>144.</math> Similarly, the area of the second largest right triangle is <math>64</math> (the side length is <math>8\sqrt{2}</math>), and the area of the overlap triangle is <math>h^2</math> (the side length is <math>h\sqrt{2}</math>) Thus,
 
<cmath>144+64-h^2=183,</cmath>
 
<cmath>144+64-h^2=183,</cmath>
 
which means that <math>\boxed{h=5}.</math>
 
which means that <math>\boxed{h=5}.</math>
  
 
~BS2012
 
~BS2012
−
 
−
~WhatdoHumanitariansEat fixed answer because BS2012 forgot to divide by 2
 
−
~changed back people are saying 5 a diagram would make it more clear
 

Revision as of 17:22, 25 January 2024

Problem

Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high while the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides form a $45^\circ$ angle with the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$

Solution 1

Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle. The side length of the largest right triangle is $12\sqrt{2},$ which means its area is $144.$ Similarly, the area of the second largest right triangle is $64$ (the side length is $8\sqrt{2}$), and the area of the overlap triangle is $h^2$ (the side length is $h\sqrt{2}$) Thus, \[144+64-h^2=183,\] which means that $\boxed{h=5}.$

~BS2012

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