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Difference between revisions of "1987 AHSME Problems/Problem 8"

(Created page with "==Problem== In the figure the sum of the distances <math>AD</math> and <math>BD</math> is <asy> draw((0,0)--(13,0)--(13,4)--(10,4)); draw((12.5,0)--(12.5,.5)--(13,.5)); draw((1...")
 
(Added a solution with explanation)
 
(One intermediate revision by one other user not shown)
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\textbf{(D)}\ \text{between 16 and 17}\qquad
 
\textbf{(D)}\ \text{between 16 and 17}\qquad
 
\textbf{(E)}\ 17</math>
 
\textbf{(E)}\ 17</math>
 +
 +
== Solution ==
 +
 +
Using Pythagoras' Theorem, <math>BD = \sqrt{3^2 + 4^2} = 5</math>, and <math>AD = \sqrt{(13 - 3)^2 + 4^2} = \sqrt{116}</math>. Thus the sum is <math>5 + \sqrt{116}</math>, and as <math>100 < 116 < 121</math>, <math>10 < \sqrt{116} < 11</math>, so that the sum is between <math>5+10</math> and <math>5+11</math>, or <math>15</math> and <math>16</math>, which is answer <math>\boxed{\text{C}}</math>.
  
 
== See also ==
 
== See also ==
−
{{AHSME box|year=1987|num-b=6|num-a=8}}   
+
{{AHSME box|year=1987|num-b=7|num-a=9}}   
  
 
[[Category: Introductory Geometry Problems]]
 
[[Category: Introductory Geometry Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 13:02, 1 March 2018

Problem

In the figure the sum of the distances $AD$ and $BD$ is

[asy] draw((0,0)--(13,0)--(13,4)--(10,4)); draw((12.5,0)--(12.5,.5)--(13,.5)); draw((13,3.5)--(12.5,3.5)--(12.5,4)); label("A", (0,0), S); label("B", (13,0), SE); label("C", (13,4), NE); label("D", (10,4), N); label("13", (6.5,0), S); label("4", (13,2), E); label("3", (11.5,4), N); [/asy]

$\textbf{(A)}\ \text{between 10 and 11} \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ \text{between 15 and 16}\qquad\\ \textbf{(D)}\ \text{between 16 and 17}\qquad \textbf{(E)}\ 17$

Solution

Using Pythagoras' Theorem, $BD = \sqrt{3^2 + 4^2} = 5$, and $AD = \sqrt{(13 - 3)^2 + 4^2} = \sqrt{116}$. Thus the sum is $5 + \sqrt{116}$, and as $100 < 116 < 121$, $10 < \sqrt{116} < 11$, so that the sum is between $5+10$ and $5+11$, or $15$ and $16$, which is answer $\boxed{\text{C}}$.

See also

1987 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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