2024 AMC 10B Problems/Problem 11

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 $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$ . Point $M$ lies $\overline{XY}$ , point $A$ lies on $\overline{YZ}$ , and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$ ?

$[asy] pair X = (0, 0); pair W = (0, 4); pair Y = (8, 0); pair Z = (8, 4); label("$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$", M, dir(-90)); [/asy]$

Note: On certain tests that took place in China, the problem asked for the area of $\triangle MAY$ .

$\textbf{(A) }13 \qquad \textbf{(B) }14 \qquad \textbf{(C) }15 \qquad \textbf{(D) }16 \qquad \textbf{(E) }17 \qquad$

Solution

Solution 1

We know that $WX = 4$ , $WZ = 8$ , so $YZ = 4$ and $YX = 8$ . Since $\angle WMA = 90^\circ$ , triangles $WXM$ and $MYA$ are similar. Therefore, $\frac{WX}{MY} = \frac{XM}{YA}$ , which gives $\frac{4}{8 - XM} = \frac{XM}{4 - ZA}$ . We also know that the areas of triangles $WXM$ and $WAZ$ are equal, so $WX \cdot XM = WZ \cdot ZA$ , which implies $4 \cdot XM = 8 \cdot ZA$ . Substituting this into the previous equation, we get $\frac{4}{8 - 2ZA} = \frac{XM}{4 - ZA}$ , yielding $ZA = 1$ and $XM = 2$ . Thus,

$\triangle WMA = 4 \cdot 8 - \frac{4 \cdot 2}{2} - \frac{8 \cdot 1}{2} - \frac{6 \cdot 3}{2} = \boxed{\textbf{(C) }15}$

~Athmyx

Solution 2

Let $XM=b$ , $ZA = a$ , $4\cdot b= 8\cdot a$ , $b = 2a$ , $WM^2 + AM^2 = AW^2$ $(b^2+4^2) + (4-a)^2 + (8-b)^2 = (a^2 + 8^2)$ a=1, b=2 , $\triangle WMA = area(WXYZ) - \triangle WZA- \triangle WXM- \triangle MYA = 32 - 4-4-9 = \boxed{\textbf{(C) }15}$

~luckuso ~minor edits by EaZ_Shadow

Solution 3 (Pythagorean Theorem)

Assign ZA as $x$ , then AY as $4 - x$ . Assign XM as $y$ and MY as $8 - y$ . Since triangles WXM and WZA are together, we can say $4x = 8y$ , so $y = 2x$ . Then therefore, XM is $2x$ and MY has length $8 - 2x$ . We can use the Pythagorean theorem to find WM, which is actually $\sqrt{(2x)^2 + 4^2)} = \sqrt{4x^2 + 16}$ . We don't factor it yet - we are going to find $x$ again using the Pythagorean Theorem. Similarly, finding MA is just the square root of the squares of AY and MY individually, or $\sqrt{(8 - 2x)^2 + (4 - x)^2} = \sqrt{64 - 32x + 4x^2 + 16 - 8x + x^2} = \sqrt{5x^2 - 40x + 80}$ . Then simply, WA is really $\sqrt{x^2 + 64}$ .

Now we have the three sides of the right triangle: $\sqrt{4x^2 + 16}$ , $\sqrt{5x^2 - 40x + 80}$ , and $\sqrt{x^2 + 64}$ . Per the Pythagorean theorem again, we can see $(4x^2 + 16) + (5x^2 - 40x + 80) = (x^2 + 64)$ . Combining like terms gives us $8x^2 - 40x + 32 = 0$ , then dividing by 8 gives $x^2 - 5x + 4 = 0$ . As this elementary and well-known quadratic gives us the roots of $1$ and $4$ , we can see it is a bit weird to have $x = 4$ , as then point Z is point A. So we'll assume $x = 1$ . We have two legs of the triangle by plugging in the sides with x in them, given that $x = 1$ : $\sqrt{20}$ and $\sqrt{45}$ . We should know that $20 \cdot 45 = 900$ , and $\sqrt{900} = 30.$ Dividing by 2 reveals us our answer: $\boxed{\textbf{(C) }15}$

~pepper2831

Solution 4 (Similar Triangles)

We are given $WX = 4$ , $WZ = 8$ . △ WXM and △ MYA have equal area, so let $XM = 2x$ and $ZA = x$ . $MY = 8-2x$ and $AY = 4-x$ . From this, we can conclude that $\frac{MY}{AY} = \frac{8-2x}{4-x} = \frac{2}{1}$

Since $WM$ intersects parallel lines $WZ$ and $XY$ , $\angle ZWM = \angle WMZ$ . $\angle ZWM + \angle MWX = 90^\circ$ , so $180^\circ - 90^\circ = \angle WMZ + \angle AMY$ . Thus, $\angle MWX = \angle AMY$ and △ WXM ~ △ MYA due to AA Similarity.

Corresponding sides of similar triangles are proportional, so $\frac{WX}{XM} = \frac{MY}{AY}$ or $\frac{4}{2x} = \frac{2}{1}$ . It is clear that $2x = 2$ , and $x = 1$ . Now, all we have to do is subtract the area of the rectangle by each of the three triangles.

△ WMA = $8$ · $4$ - ( $\frac{1}{2}$ · $4$ · $2$ ) - ( $\frac{1}{2}$ · $8$ · $1$ ) - ( $\frac{1}{2}$ · $6$ · $3$ )

△ WMA = $32 - 4 - 4 - 9$

△ WMA = $\boxed{\textbf{(C) }15}$

~peeghj

China Test Solution (Finding MAY)

From solution 3, instead of finding $WMA$ , we instead find MAY. Then $x = 1$ then we have $MA = 8 - 2x = 6$ . Again, since $AY = 4 - x$ , then $AY = 4 - 1 = 3.$ The area of a triangle with legs $3$ and $6$ is $\frac{3 * 6}{2} = \boxed{9}$ .

~pepper2831 (again)

Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)

https://youtu.be/YqKmvSR1Ckk?feature=shared

~ Pi Academy

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

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

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