Geometry Selected AMC 10 Practice Problems
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Question 1 of 4
1. Question
[AMC 10] What is the volume of tetrahedron \(ABCD\) with edge lengths \(AB = 2, AC = 3, AD = 4, BC = \sqrt{13}, BD = 2\sqrt{5}, \text{ and } CD = 5\) ?
\(\textbf{(A)} ~3 \qquad\textbf{(B)} ~2\sqrt{3} \qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~3\sqrt{3}\qquad\textbf{(E)} ~6\)
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Question 2 of 4
2. Question
[AMC 10] Trapezoid
has 
, and 
. Let 
be the intersection of the diagonals 
and 
, and let 
be the midpoint of . Given that 
, the length of 
can be written in the form 
, where 
and 
are positive integers and is not divisible by the square of any prime. What is 
?
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Question 3 of 4
3. Question
[AMC 10] Square \(ABCD\) has side length \(2\). A semicircle with diameter \(\overline{AB}\) is constructed inside the square, and the tangent to the semicircle from \(C\) intersects side \(\overline{AD}\) at \(E\). What is the length of \(\overline{CE}\)?
![[asy] size(100); defaultpen(fontsize(10)); pair A=(0,0), B=(2,0), C=(2,2), D=(0,2), E=(0,1/2); draw(A--B--C--D--cycle);draw(C--E); draw(Arc((1,0),1,0,180)); label("$A$",A,(-1,-1)); label("$B$",B,( 1,-1)); label("$C$",C,( 1, 1)); label("$D$",D,(-1, 1)); label("$E$",E,(-1, 0)); [/asy]](https://latex.artofproblemsolving.com/d/a/e/daef6f3ebe3e1fc6cfacd8611db1d6d449c8d2d4.png)
\( \mathrm{(A) \ } \frac{2+\sqrt{5}}{2} \qquad \mathrm{(B) \ } \sqrt{5} \qquad \mathrm{(C) \ } \sqrt{6} \qquad \mathrm{(D) \ } \frac{5}{2} \qquad \mathrm{(E) \ } 5-\sqrt{5} \)
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Question 4 of 4
4. Question
[AIME] Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle \(C\) with radius 30. Let \(\) be the area of the region inside circle \(C\) and outside of the six circles in the ring. Find \( \lfloor K \rfloor\) (the floor function).
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