Difference between revisions of "2009 AMC 8 Problems/Problem 7"

m (Problem)
(Solution)
 
(One intermediate revision by one other user not shown)
Line 2: Line 2:
  
 
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad.  Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles.  The width of the railroad track can be ignored.  How many square miles are in the plot of land ACD?
 
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad.  Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles.  The width of the railroad track can be ignored.  How many square miles are in the plot of land ACD?
 +
 
<asy>
 
<asy>
 
size(250);
 
size(250);
Line 35: Line 36:
 
label(rotate(63.43494882)*"Brown", A--D, NW);
 
label(rotate(63.43494882)*"Brown", A--D, NW);
 
</asy>
 
</asy>
<math>\textbf{(A)}\ 2\qquad
+
 
\textbf{(B)}\ 3 \qquad
+
<math>\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9</math>
\textbf{(C)}\ 4.5 \qquad
 
\textbf{(D)}\ 6 \qquad
 
\textbf{(E)}\ 9</math>
 
  
 
==Solution==
 
==Solution==
 
The area of a triangle is <math>\frac12 bh</math>. If we let <math>CD</math> be the base of the triangle, then the height is <math>AB</math>, and the area is <math>\frac12 \cdot 3 \cdot 3 = \boxed{\textbf{(C)}\ 4.5}</math>.
 
The area of a triangle is <math>\frac12 bh</math>. If we let <math>CD</math> be the base of the triangle, then the height is <math>AB</math>, and the area is <math>\frac12 \cdot 3 \cdot 3 = \boxed{\textbf{(C)}\ 4.5}</math>.
 +
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=Opz71P5o4uI
  
 
==See Also==
 
==See Also==
 
{{AMC8 box|year=2009|num-b=6|num-a=8}}
 
{{AMC8 box|year=2009|num-b=6|num-a=8}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:22, 10 June 2022

Problem

The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD?

[asy] size(250); defaultpen(linewidth(0.55)); pair A=(-6,0), B=origin, C=(0,6), D=(0,12); pair ac=C+2.828*dir(45), ca=A+2.828*dir(225), ad=D+2.828*dir(A--D), da=A+2.828*dir(D--A), ab=(2.828,0), ba=(-6-2.828, 0);  fill(A--C--D--cycle, gray); draw(ba--ab); draw(ac--ca); draw(ad--da); draw((0,-1)--(0,15)); draw((1/3, -1)--(1/3, 15)); int i; for(i=1; i<15; i=i+1) { draw((-1/10, i)--(13/30, i)); }  label("$A$", A, SE); label("$B$", B, SE); label("$C$", C, SE); label("$D$", D, SE); label("$3$", (1/3,3), E); label("$3$", (1/3,9), E); label("$3$", (-3,0), S); label("Main", (-3,0), N); label(rotate(45)*"Aspen", A--C, SE); label(rotate(63.43494882)*"Brown", A--D, NW); [/asy]

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$

Solution

The area of a triangle is $\frac12 bh$. If we let $CD$ be the base of the triangle, then the height is $AB$, and the area is $\frac12 \cdot 3 \cdot 3 = \boxed{\textbf{(C)}\ 4.5}$.

Video Solution

https://www.youtube.com/watch?v=Opz71P5o4uI

See Also

2009 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Invalid username
Login to AoPS