Difference between revisions of "2024 AMC 10B Problems/Problem 11"
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==Problem== | ==Problem== | ||
| + | In the figure below <math>WXYZ</math> is a rectangle with <math>WX=4</math> and <math>WZ=8</math>. Point <math>M</math> lies <math>\overline{XY}</math>, point <math>A</math> lies on <math>\overline{YZ}</math>, and <math>\angle WMA</math> is a right angle. The areas of <math>\triangle WXM</math> and <math>\triangle WAZ</math> are equal. What is the area of <math>\triangle WMA</math>? | ||
| + | |||
| + | <asy> | ||
| + | pair X = (0, 0); | ||
| + | pair W = (0, 4); | ||
| + | pair Y = (8, 0); | ||
| + | pair Z = (8, 4); | ||
| + | label("$X$", X, dir(180)); | ||
| + | label("$W$", W, dir(180)); | ||
| + | label("$Y$", Y, dir(0)); | ||
| + | label("$Z$", Z, dir(0)); | ||
| + | |||
| + | draw(W--X--Y--Z--cycle); | ||
| + | dot(X); | ||
| + | dot(Y); | ||
| + | dot(W); | ||
| + | dot(Z); | ||
| + | pair M = (2, 0); | ||
| + | pair A = (8, 3); | ||
| + | label("$A$", A, dir(0)); | ||
| + | dot(M); | ||
| + | dot(A); | ||
| + | draw(W--M--A--cycle); | ||
| + | markscalefactor = 0.05; | ||
| + | draw(rightanglemark(W, M, A)); | ||
| + | label("$M$", M, dir(-90)); | ||
| + | </asy> | ||
| + | |||
| + | <math> | ||
| + | \textbf{(A) }13 \qquad | ||
| + | \textbf{(B) }14 \qquad | ||
| + | \textbf{(C) }15 \qquad | ||
| + | \textbf{(D) }16 \qquad | ||
| + | \textbf{(E) }17 \qquad | ||
| + | </math> | ||
| + | |||
| + | [[2024 AMC 12B Problems/Problem 7|Solution]] | ||
==Solution 1== | ==Solution 1== | ||
Revision as of 05:38, 14 November 2024
- The following problem is from both the 2024 AMC 10B #11 and 2024 AMC 12B #7, so both problems redirect to this page.
Problem
In the figure below 
is a rectangle with 
and 
. Point 
lies 
, point 
lies on 
, and 
is a right angle. The areas of 
and 
are equal. What is the area of 
?
![[asy] pair X = (0, 0); pair W = (0, 4); pair Y = (8, 0); pair Z = (8, 4); label("$X$", X, dir(180)); label("$W$", W, dir(180)); label("$Y$", Y, dir(0)); label("$Z$", Z, dir(0)); draw(W--X--Y--Z--cycle); dot(X); dot(Y); dot(W); dot(Z); pair M = (2, 0); pair A = (8, 3); label("$A$", A, dir(0)); dot(M); dot(A); draw(W--M--A--cycle); markscalefactor = 0.05; draw(rightanglemark(W, M, A)); label("$M$", M, dir(-90)); [/asy]](http://latex.artofproblemsolving.com/a/e/9/ae9f151d0a601f719d48015b0f36651f319802a3.png)
Solution 1
See also
| 2024 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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 AMC 10 Problems and Solutions | ||
| 2024 AMC 12B (Problems • Answer Key • Resources) | |
| 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 AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
