How many non-congruent right triangles are there such that the perimeter in \(\text{cm}\) and the area in \(\text{cm}^2\) are numerically equal? $$\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ \text{infinitely many}$$
Tags: geometry,area,perimeter,right-triangle,area-of-triangle,triangle-inequality,trigonometry,pythagoras-theorem
Source: AMC 12 1983 A Q24
Elo Rating: 1484.00
A circle with center \( C\) is tangent to the positive \( x\) and \( y\)-axes and externally tangent to the circle centered at \( (3,0)\) with radius \( 1\). What is the sum of all possible radii of the circle with center \( C\)? $$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$$
Tags: circles,tangent,tangent-cirlces,coordinate-geometry,geometry
Source: AMC 12 2009 A Q16
Elo Rating: 1500.00