Geometry

 

1. Similar Triangles in a Complex Figure
In the figure below, $E$ and $G$ lie on $\overline{AC}$, $D$ and $F$ lie on $\overline{AB}$, and $\overline{DE}$ and $\overline{FG}$ are parallel to $\overline{BC}$. If the given segment lengths are $DE=7$, $FG=6$, $BC=16$, and $AB=8$, what is the length of $AC$?
(A) 13
(B) 26
(C) 29
(D) 42
(E) 48

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2. Midpoint in the Coordinate Plane
In the standard $(x,y)$ coordinate plane, the point $(2,1)$ is the midpoint of segment $\overline{CD}$. Point $C$ has coordinates $(6,8)$. What are the coordinates of point $D$?
(F) $(-2,-6)$
(G) $(10,15)$
(H) $(10,10)$
(J) $(1,4)$
(K) $(1,-2)$

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3. Triangle Angle Calculation
In triangle $ABC$, the measure of angle $A$ is ${58}^{\circ }$, and $\overline{AB}\cong \overline{AC}$. What is the measure of angle $C$?
(A) 32°
(B) 42°
(C) 58°
(D) 61°
(E) 62°

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4. Equation of a Circle
In the standard $(x,y)$ coordinate plane, a circle with center at $(8,5)$ and radius 9 units has which equation?
(F) $(x-8{)}^2+(y-5{)}^2=81$
(G) $(x-8{)}^2+(y-5{)}^2=9$
(H) $(x+8{)}^2+(y+5{)}^2=81$
(J) $(x+8{)}^2+(y+5{)}^2=9$
(K) $(x+5{)}^2+(y+8{)}^2=81$

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5. Area of a Circle from Circumference
What is the area, in square units, of a circle that has a circumference of $12\pi$ units?
(A) $6\pi$
(B) $12\pi$
(C) $24\pi$
(D) $36\pi$
(E) $144\pi$

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6. Rectangle Area and Dimensions
One side of square $ABCD$ has length 12 meters. A rectangle has the same area as $ABCD$ and a width of 8 meters. What is the length of the rectangle?
(A) 12
(B) 16
(C) 18
(D) 20
(E) 24

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7. Slope of a Parallel Line
What is the slope of a line, in the standard $(x,y)$ coordinate plane, that is parallel to $x+5y=9$?
(F) $-5$
(G) $-1\mathrm{/}5$
(H) $1\mathrm{/}5$
(J) $5$
(K) $9$

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8. Properties of Special Triangles
In a circle centered at $O$, points $A,B,C,D$ lie on the circle, and triangle $ABC$ is shown with $\mathrm{\angle }A={60}^{\circ }$. Which of the following does NOT appear in the figure?
(A) Acute triangle
(B) Equilateral triangle
(C) Isosceles triangle
(D) Right triangle
(E) Scalene triangle

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9. Volume/Area of a Composite Solid
A solid rectangular prism is built by alternating congruent black cubes and white cubes such that no two cubes of the same color share more than one edge. If the prism is $7\times 5\times 6$ cubes, what is the total number of white cubes used?
(F) 45
(G) 102
(H) 105
(J) 140
(K) 210

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10. Area of a Quadrilateral in Terms of Variables
Points $M(2a,5b),N(4a,9b),P(6a,5b)$ are on a parabola, and $Q(4a,0)$ is not. What is the area, in square coordinate units, of quadrilateral $MPNQ$?
(F) $8ab$
(G) $10ab$
(H) $12ab$
(J) $15ab$
(K) $20ab$