Graduate Record Examinations (GRE) Quantitative Reasoning: Geometry Flashcards

24 sample cards from the Quantitative Reasoning: Geometry deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What is the relationship between a central angle and the inscribed angle subtending the same arc?

   The central angle is twice the inscribed angle that subtends the same arc: $\theta_{\text{central}} = 2\,\theta_{\text{inscribed}}$.

2. What is the measure of an inscribed angle that subtends a diameter (semicircle)?

   $90^{\circ}$. An angle inscribed in a semicircle is always a right angle (Thales' theorem).

3. How do you find the arc length of a sector with central angle $\theta$ (in degrees) in a circle of radius $r$?

   $$\text{Arc length} = \frac{\theta}{360^{\circ}}\times 2\pi r$$

4. What is the area of a sector with central angle $\theta$ (in degrees) and radius $r$?

   $$A = \frac{\theta}{360^{\circ}}\times \pi r^{2}$$

5. What is the relationship between a radius drawn to the point of tangency and the tangent line?

   The radius is perpendicular to the tangent line at the point of tangency, forming a $90^{\circ}$ angle.

6. What is the general strategy for finding the area of a shaded region in a composite figure?

   Find the area of the larger/outer figure and subtract the area of the unshaded inner figure(s): Shaded $=$ Total area $-$ Unshaded area.

7. How do you find the area of a composite figure made of several simple shapes?

   Decompose it into non-overlapping basic shapes (rectangles, triangles, semicircles), compute each area, then add them together.

8. What is the surface area of a rectangular solid (box) with dimensions $l$, $w$, and $h$?

   $$SA = 2(lw + lh + wh)$$

9. What is the volume of a rectangular solid with dimensions $l$, $w$, and $h$?

   $$V = lwh$$

10. Give the formulas for the volume and surface area of a sphere of radius $r$.

    Volume $V = \frac{4}{3}\pi r^{3}$ and surface area $SA = 4\pi r^{2}$.

11. What are the volume and lateral (curved) surface area of a right circular cylinder with radius $r$ and height $h$?

    Volume $V = \pi r^{2}h$; lateral surface area $= 2\pi r h$; total surface area $= 2\pi r h + 2\pi r^{2}$.

12. What are the volume and lateral surface area of a right circular cone with radius $r$, height $h$, and slant height $l$?

    Volume $V = \frac{1}{3}\pi r^{2}h$; lateral surface area $= \pi r l$, where $l = \sqrt{r^{2}+h^{2}}$.

13. What is the formula for the length of the space diagonal of a rectangular box with dimensions $l$, $w$, $h$?

    $$d = \sqrt{l^{2} + w^{2} + h^{2}}$$

14. What is the length of the space diagonal of a cube with edge length $s$?

    $$d = s\sqrt{3}$$

15. What is the length of the face diagonal of a cube with edge length $s$?

    $$d = s\sqrt{2}$$

16. When a cube is unfolded into a net, how many faces, edges, and vertices does the cube have?

    A cube has 6 (square) faces, 12 edges, and 8 vertices. A valid net shows all 6 squares connected so they fold without overlap.

17. What does Euler's formula state for the faces, vertices, and edges of a convex polyhedron?

    $$V - E + F = 2$$ where $V$ = vertices, $E$ = edges, $F$ = faces.

18. State the distance formula between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane.

    $$d = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2}}$$

19. State the midpoint formula for the segment joining $(x_1, y_1)$ and $(x_2, y_2)$.

    $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

20. What is the standard equation of a circle with center $(h, k)$ and radius $r$?

    $$(x - h)^{2} + (y - k)^{2} = r^{2}$$

21. How do you find the slope of the line through points $(x_1, y_1)$ and $(x_2, y_2)$?

    $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

22. What is the slope-intercept form of a line, and what does each variable represent?

    $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept (the value of $y$ where the line crosses the $y$-axis).

23. What is the relationship between the slopes of two parallel lines versus two perpendicular lines?

    Parallel lines have equal slopes ($m_1 = m_2$); perpendicular lines have slopes that are negative reciprocals ($m_1 \cdot m_2 = -1$).

24. How do you find the $x$-intercept and $y$-intercept of a line from its equation?

    For the $x$-intercept, set $y = 0$ and solve for $x$; for the $y$-intercept, set $x = 0$ and solve for $y$.

What this deck covers

Quantitative Reasoning: Geometry flashcards FAQ