Geometry and Trigonometry: Lines, Angles, and Triangles

Practice Questions

Level 1: Easy (Foundational)

1. A satellite tracks an aircraft in 3D coordinate space. At time $t=1$, the aircraft is at position $A(2, -4, 5)$ and at time $t=3$ it is at $B(8, 6, -1)$. What are the coordinates of the midpoint of the aircraft's path between these two times?

   A) $(3, 1, 2)$
   B) $(5, 1, 2)$
   C) $(5, 2, 3)$
   D) $(6, 10, -6)$

2. A submarine's guidance system uses column vectors to plot courses. The current velocity vector is $\mathbf{u} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}$ and the ocean current vector is $\mathbf{v} = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}$. What is the resultant vector of $2\mathbf{u} + \mathbf{v}$?

   A) $\begin{pmatrix} 5 \\ 0 \\ 4 \end{pmatrix}$
   B) $\begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}$
   C) $\begin{pmatrix} 7 \\ -8 \\ 0 \end{pmatrix}$
   D) $\begin{pmatrix} 5 \\ 8 \\ 4 \end{pmatrix}$

3. [Student-Produced Response] A drone takes off from a launchpad at the origin $(0,0,0)$ and hovers exactly at the 3D coordinates $(2, 3, 6)$, where all units are in meters. What is the exact straight-line distance, in meters, from the launchpad to the drone?

4. An architect is designing a monument in the shape of a right pyramid with a square base. If the side length of the square base is $6$ meters and the vertical height of the pyramid is $10$ meters, what is the exact volume of the monument in cubic meters?

   A) $60$
   B) $120$
   C) $180$
   D) $360$

5. In a video game, the movement of a character is defined by the vector $\mathbf{a} = \begin{pmatrix} 4 \\ k \end{pmatrix}$, and a laser beam travels along the vector $\mathbf{b} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$. If the character's path is perfectly perpendicular to the laser beam, what is the value of the constant $k$?

   A) $-6$
   B) $-1.5$
   C) $1.5$
   D) $6$

Level 2: Medium (Standard)

1. Determine the exact area of a triangular garden plot that has two sides measuring $8$ meters and $10$ meters, with an included angle of $150^\circ$ between them.

   A) $20$
   B) $20\sqrt{3}$
   C) $40$
   D) $40\sqrt{3}$

2. A surveyor measures a triangular parcel of land. Two sides measure $3$ kilometers and $8$ kilometers, and the angle between these two sides is $60^\circ$. What is the exact length of the third side of the parcel in kilometers?

   A) $7$
   B) $8$
   C) $\sqrt{73}$
   D) $\sqrt{97}$

3. [Student-Produced Response] A drone moves at a constant velocity, and its position vector $\mathbf{r}$ at time $t$ seconds is given by the kinematic equation $\mathbf{r} = \begin{pmatrix} 2 \\ 5 \end{pmatrix} + t \begin{pmatrix} 3 \\ -4 \end{pmatrix}$ in meters. What is the exact speed of the drone in meters per second?

4. In a Voronoi diagram mapping out competing delivery zones, the boundary between Store A at $(1, 1)$ and Store B at $(5, 5)$ is the perpendicular bisector of the line segment connecting them. What is the equation of this boundary line?

   A) $y = x$
   B) $y = -x + 6$
   C) $y = -x + 8$
   D) $y = 0.5x + 1.5$

5. [Student-Produced Response] A paperweight is shaped like a solid hemisphere with a radius of $4$ centimeters. The total surface area of the paperweight, including its flat circular base, is expressed as $k\pi$ square centimeters. What is the value of $k$?

Level 3: Hard (Advanced)

1. Two aircraft are flying along straight-line paths described by vector equations. Aircraft 1 has position $\mathbf{r}_1 = \begin{pmatrix} 0 \\ 10 \end{pmatrix} + t \begin{pmatrix} 4 \\ -2 \end{pmatrix}$ and Aircraft 2 has position $\mathbf{r}_2 = \begin{pmatrix} 12 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ c \end{pmatrix}$, where $t$ is time in hours. If the two aircraft collide (reach the same position at the exact same time), what must be the value of the constant $c$?

   A) $-1$
   B) $-0.5$
   C) $0.5$
   D) $1$

2. Investigate the geometric relationship between the 3D vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$. By computing their scalar product, determine the exact cosine of the angle $\theta$ between them.

   A) $\frac{8}{9}$
   B) $\frac{7}{9}$
   C) $\frac{2}{3}$
   D) $\frac{\sqrt{5}}{3}$

3. In triangle $ABC$, side $a = 10$, side $b = 10\sqrt{3}$, and angle $A = 30^\circ$. Explain why the ambiguous case occurs, and determine the two possible mathematical values for angle $B$.

   A) $30^\circ$ and $150^\circ$
   B) $60^\circ$ and $120^\circ$
   C) $45^\circ$ and $135^\circ$
   D) Only $60^\circ$ is possible.

4. A glass paperweight is a solid square-based pyramid. The side length of the square base is $10$ cm, and the vertical height of the pyramid (from the center of the base to the apex) is $12$ cm. Show that the slant height of a triangular face is $13$ cm, and determine the exact length of one of the slanted edges connecting a base corner to the apex.

   A) $\sqrt{169}$
   B) $\sqrt{194}$
   C) $\sqrt{269}$
   D) $15$

5. [Student-Produced Response] A high-speed particle moves through a 3D magnetic field along a linear path defined by the vector equation $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$. At a certain instant, the particle passes through a sensor located at a point $P$ on this line. If the $y$-coordinate of point $P$ is $-5$, what is the exact $z$-coordinate of point $P$?