Difference between revisions of "2021 AMC 12A Problems/Problem 2"

Revision as of 15:32, 11 February 2021

Problem

Under what conditions does $\sqrt{a^2+b^2}=a+b$ hold, 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

Square both sides to get $a^{2}+b^{2}=a^{2}+2ab+b^{2}$ . Then, $0=2ab\rightarrow ab=0$ . Then, the answer is $\boxed{\textbf{(B)}}$ . Consider a right triangle with legs $a$ and $b$ and hypotenuse $\sqrt{a^{2}+b^{2}}$ . Then one of the legs must be equal to $0$ , but they are also nonnegative as they are lengths. Therefore, both $\textbf{(B)}$ and $\textbf{(D)}$ are correct.

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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.

@@ Line 1: / Line 1: @@
 ==Problem==
-Under what conditions is <math>\sqrt{a^2+b^2}=a+b</math> true, where <math>a</math> and <math>b</math> are real numbers?
+Under what conditions does <math>\sqrt{a^2+b^2}=a+b</math> hold, where <math>a</math> and <math>b</math> are real numbers?
 <math>\textbf{(A) }</math> It is never true.

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Difference between revisions of "2021 AMC 12A Problems/Problem 2"

Revision as of 15:32, 11 February 2021

Problem

Solution

See also