2021 AMC 12A Problems/Problem 2


Under what conditions is $\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?

$\textbf{(A) }$ It is never true.

$\textbf{(B) }$ It is true if and only if $ab=0$.

$\textbf{(C) }$ It is true if and only if $a+b\ge 0$.

$\textbf{(D) }$ It is true if and only if $ab=0$ and $a+b\ge 0$.

$\textbf{(E) }$ It is always true.

Solution 1 (Algebra)

One can square both sides to get $a^{2}+b^{2}=a^{2}+2ab+b^{2}$. Then, $0=2ab\implies ab=0$. Also, it is clear that both sides of the equation must be nonnegative. The answer is $\boxed{\textbf{(D)}}$.


Solution 2 (Algebra)

Complete the square of the left side by rewriting the radical to be \[\sqrt{(a+b)^{2}-2ab}.\] From there it is evident for the square root of the left to be equal to the right, $ab$ must be equal to zero. Also, we know that the equivalency of square root values only holds true for nonnegative values of $a+b$, making the correct answer $\boxed{\textbf{(D)}}.$


Solution 3 (Process of Elimination)

The left side of the original equation is the arithmetic square root, which is always nonnegative. So, we need $a+b\ge 0,$ which refutes $\textbf{(B)}$ and $\textbf{(E)}.$ Next, picking $(a,b)=(0,0)$ refutes $\textbf{(A)},$ and picking $(a,b)=(1,2)$ refutes $\textbf{(C)}.$ By POE (Process of Elimination), the answer is $\boxed{\textbf{(D)}}.$


Solution 4 (Graphing)

If we graph $\sqrt{x^2+y^2}=x+y,$ then we get the union of:

  • positive $x$-axis
  • positive $y$-axis
  • origin

Therefore, the answer is $\boxed{\textbf{(D)}}.$

The graph of $\sqrt{x^2+y^2}=x+y$ is shown below. [asy] /* Made by MRENTHUSIASM */ size(200);   int xMin = -5; int xMax = 5; int yMin = -5; int yMax = 5;  draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2)); draw((xMax,0)--(0,0)--(0,yMax),red+linewidth(1.5)); [/asy] ~MRENTHUSIASM (credit given to TheAMCHub)

Video Solution (Quick and Easy)


~Education, the Study of Everything

Video Solution by Aaron He


Video Solution by Hawk Math


Video Solution by OmegaLearn (Using Logic and Analyzing Answer Choices)


~ pi_is_3.14

Video Solution by TheBeautyofMath



See also

2021 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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All AMC 12 Problems and Solutions

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