2012 AMC 8 Problems/Problem 25

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A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$?

[asy] draw((0,2)--(2,2)--(2,0)--(0,0)--cycle); draw((0,0.3)--(0.3,2)--(2,1.7)--(1.7,0)--cycle); label("$a$",(-0.1,0.15)); label("$b$",(-0.1,1.15));[/asy]


Solution 1

The total area of the four congruent triangles formed by the squares is $5-4 = 1$. Therefore, the area of one of these triangles is $\frac{1}{4}$. The height of one of these triangles is $a$ and the base is $b$. Using the formula for area of the triangle, we have $\frac{ab}{2} = \frac{1}{4}$. Multiply by $2$ on both sides to find that the value of $ab$ is $\boxed{\textbf{(C)}\ \frac{1}2}$.

Solution 2

To solve this problem you could also use algebraic manipulation.

Since the area of the large square is $5$, the side length is $\sqrt{5}$.

We then have the equation $a + b = \sqrt{5}$.

We also know that the side length of the smaller square is $2$, since its area is $4$. Then, the segment of length $a$ and segment of length $b$ form a right triangle whose hypotenuse would have length $2$.

So our second equation is $\sqrt{{a^2}+{b^2}} = 2$.

Square both equations.

$a^2 + 2ab + b^2 = 5$

$a^2 + b^2 = 4$

Now, subtract, and obtain the equation $2ab = 1$. We can deduce that the value of $a \cdot b$ is $\boxed{\textbf{(C)}\ \frac{1}2}$.

Solution 3 (similar to solution 1)

Since we know that each of the $4$ triangles have side lengths $a$ and $b$, we can create an equation: the area of the inner square plus the sum of the $4$ triangles equals the area of the outer square.

\[4 + 2ab = 5\]

which gives us the value of $a \cdot b$, which is $\boxed{\textbf{(C)}\ \frac{1}2}$.

Solution 4

First, observe that the given squares have areas $4$ and $5$.

Then, observe that the 4 triangles with side lengths $a$ and $b$ have a combined area of $5-4=1$.

We have, that $4\cdot\frac{ab}{2}=2ab$ is the total area of the 4 triangles in terms of $a$ and $b$.

Since $2ab=1$, we divide by two getting $a \cdot b=\boxed{\textbf{(C) }\frac{1}{2}}$

Video Solution by Punxsutawney Phil



https://www.youtube.com/watch?v=QEwZ_17PQ6Q ~David

Video Solution 2

https://youtu.be/MhxGq1sSA6U ~savannahsolver

Video Solution by OmegaLearn


~ pi_is_3.14

See Also

2012 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 24
Followed by
Last Problem
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All AJHSME/AMC 8 Problems and Solutions

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