Geometry Flashcards
(45 cards)
1
Q
Transversal
A
A line that passes through two or more lines at different points
2
Q
Triangle: Angles and Sides
A
- The largest angle is always the opposite the longest side of the triangle
- The smallest angle is always the opposite the shortest side of the triangle
- Equal sides will always be opposite equal angles
3
Q
Triangle Inequality Theorem
A
- In any triangle, the sum of the lengths of any 2 sides of the triangle is greater than the 3rd side.
- In any triangle, the difference of the lengths of any 2 sides of the triangle is less than the 3rd side.
4
Q
Acute Angle vs. Right Angle
A
- When a2 + b2 = c2 → right triangle
- When a2 + b2 > c2 → acute triangle
5
Q
Pythagorean Triplet
A
- 3-4-5
- 5-12-13
- 6-8-10
- 7-24-25
- 8-15-17
- 9-12-15
- 12-16-20
6
Q
Equilateral Triangle Area
A
s2√3 / 4
7
Q
Similar Triangles
A
Two or more triangles are similar if they have the same shape. 3 ways to be similar:
- 3 angles of one triangle are the same measure as 3 angles of another triangle.
- 3 pairs of corresponding sides have lengths in the same ratio.
- An angle of one triangle is the same measure as an angle of another triangle and the sides surrounding these angles are in the same ratio.
8
Q
Congruent Triangles
A
Congruent triangles have both the same shape and same size.
9
Q
Parallelograms
A
- All rectangles are parallelograms
- The diagonals of a parallelogram bisect each other
10
Q
Max Area of a Rectangle
A
Given a rectangle with a fixed perimeter, the rectangle with the maximum area is a square.
11
Q
Min Perimeter of a Rectangle
A
Given a rectangle with a fixed area, the rectangle with the min perimeter is a square.
12
Q
Polygon Interior Angles
A
(n - 2) x 180
n = # of sides
13
Q
Polygon Exterior Angles
A
Given any polygon, when taking one exterior angle at each vertex, the sum of the measures of the exterior angles will always equal 360º.
14
Q
Regular Polygon
A
- Regular polygon = all interior angles have equal measure and all sides are equal in length
- Measure of one interior angle in a regular polygon = 180(n - 2) / n
15
Q
Hexagon Area
A
- Formula 1: (3√3/2)s2 ≈ 2.6s2
- Formula 2: ½ ap, where a = apothem (perpendicular distance from the center to a side) and p = perimeter
- Formula 3: 3/2ds, where d = distance between any two parallel sides (d = 2a), s = length of a side (p = 6s)
Formula 2 and Formula 3 are the same
16
Q
Rhombus: Area
A
#1: Base x Height (parallelogram) #2: (D<sub>1</sub> x D<sub>2</sub>) ÷ 2 (half of rectangle)
- Rhombus is a quadrilateral whose 4 sides all have the same length.
- A rhombus is a parallelogram.
- The diagonals bisect each other at right angles.
17
Q
Rectangular Solid: Diagonal
A
D = √(L2 + W2 + H2)
18
Q
Triangle Circle Intersection
A
Can intersect at 1,2,3,4,5,6 points
19
Q
Circle: Tangent
A
A line tangent to a circle forms a right angle.
20
Q
Circle:
3 Equivalent Ratios
A
In any circle, the following ratios are equal:
Central Angle / 360 = Arc Length / Circumference = Area of Sector / Area of Circle
21
Q
Circle:
Minor Arc vs. Major Arc
A
If points A and B are two points on a circle and arc AB is not a semicircle, arc AB refers to the shorter portion of the circumference between A and B.
- Minor arc: The shorter portion
- Major arc: The longer portion (usually referred as arc ACB, C representing some third point on this longer portion of the circumference)
22
Q
Circle:
Inscribed Angle vs. Central Angle
A
- The degree measure of an inscribed angle is equal to half of the degree measure of the arc that it intercepts
- When an inscribed angle shares the same endpoints as the central angle, the degree measure of the central angle is twice the degree measure of the inscribed angle
23
Q
Inscribed Angle:
Triangle Inscribed in a Circle
A
When a triangle inscribed in a circle, if one side of the triangle is also the diameter of the circle, then the triangle is a right triangle with the 90º angle opposite the diameter.
The hypotenuse of the right triangle is a diameter of the circle.
24
Q
Inscribed Angle:
Equilateral Triangle Inscribed in Circles & Angles
A
When an equilateral triangle is inscribed in a circle, the center of the triangle coincides with the center of the circle.
If we were to draw a line segment from this center to a vertex on the triangle, not only would that line segment be a radius of the circle, but it would also bisect the 60º angle.
25
Inscribed Angle:
Equilateral Triangle Inscribed in Circles & Arcs
When an equilateral triangle is inscribed in a circle, the triangle divides the circumference of the circle into 3 arcs of equal length.
26
Inscribed Angle:
Circle Inscribed in Equilateral Triangles
A circle is inscribed in an equilateral triangle when:
1. each side of the equilateral triangle is tangent to the circle,
2. the circle is the largest circle that can fit inside the equilateral triangle, and
3. the two figures touch one another at exactly 3 points.
Each point at which the circle touches the triangle is the midpoint of a side of the triangle.
27
Inscribed Angle:
Squares Inscribed in Circles
When a square is inscribed in a circle (when all of the square's vertices lie on the circumference of the circle), a diagonal of the square is also a diameter of the circle.
28
Inscribed Angle:
Circles Inscribed in Squares
When a circle is inscribed in a square (when each side of the square is tangent to the circle), the circle's diameter = square's side.
Each point at which the circle touches the square is the midpoint of a side of the square.
29
Inscribed Angle:
Triangles Inscribed in a Square
When a triangle is inscribed in a square, one side of the triangle will coincide with one side of the square. When this happens, that side of the triangle = that side of the square.
30
Inscribed Angle:
Rectangles Inscribed in Circles
When a rectangle is inscribed in a circle (when all 4 of the rectangle's vertices lie on the circumference of the circle), a diagonal of the rectangle = diameter of the circle
31
Inscribed Angle:
Regular Polygons Inscribed in Circles
When a regular polygon is inscribed in a circle, the polygon divides the circle into arcs of equal length.
32
Inscribed Angle:
Inscribing a Square Within a Square
The area of an inscribed square will be **smallest** when the vertices of that square are located at the **midpoints** of the respective edges of the circumscribed square. Furthermore, the area of such an inscribed square will be **half** of the area of the circumscribed square.
33
Inscribed Angle:
Rectangles Inscribed in Semicircles
If the length of the rectangle is x, then the base of the triangle is ½x. If the width of the rectangle is y, then the height of the triangle is y. The hypotenuse of the triangle is also the radius of the semicircle.
34
Inscribed Angle:
Square Inscribed in a Semicircle
When a square is inscribed in a semicircle and a radius is drawn from one of the vertices of the square on the semicircular arc to the center of the circle, a right triangle is created in which:
* the base of the triangle is equal to half of the side of the square,
* the height is equal to the side of the square, and
* the hypotenuse is equal to the radius of the circle.
35
Rectangular Uniform Border
A “uniform border” is a border of equal width that surrounds some object.
BE CAREFUL: The area of the frame is NOT the area of the big outer rectangle, just the frame portion!
36
Circular Ring Area
Area of the outer ring = π (R22 - R12)
37
Diagonal of Rectangular Solid vs. Square
* Given any rectangular solid or cube, the longest line segment that can be drawn within the solid will be a **diagonal** that goes from a corner of the box or cube, through the center of the box or cube, to the opposite corner.
→ This means we need to know the **length, height, and width** of the solid to calculate.
* **Rectangular solid diagonal**: D2 = W2 + L2 + H2
* **Cube diagonal**: D2 = 3x2 (x = length of one side)
38
Cylinder
* There are various types of cylinders: circular vs. elliptic, right vs. oblique
* GMAT always refers to right circular cylinders
* **Right circular cylinder**: Base are circles that are perpendicular to the height of the cylinder
39
Cylinder:
Volume and Surface Area
Volume = πr2h
Surface Area = 2πr2 + 2πrh
40
Cylinder and Rectangle Problem
Some glass vases in the shape of a right circular cylinder are to be shipped. For shipment they are to be packed into rectangular boxes. To ensure no damage, the vase must sit upright in the box and fit snugly to the top, bottom, and two opposing sides of the box. If the shipping box has dimensions of 6 inches by 8 inches by 10 inches, what is the minimum volume of a vase that could be packed in such a box?
* Answer = 72π
* There are 3 rectangular base scenarios: 6 x 8, 8 x 10, and 6 x 10
* The diameter of the cylinder has to be equal to the shortest side to fit → shortest diameter = 6 (i.e., r = 3)
* The 6 x 10 base scenario (vs. the 6 x 8 base scenario) involves the shortest height (8), so volume = π(3)2(8) = 72π
41
Volume Rate Trap
* **Liquid height**: To determine the rate at which a liquid will rise within a 3-dimensional object, we must know the rate at which the liquid flows into the figure and the _exact dimensions_ (L x W x H) of the figure.
* **Placing objects in a box**: To determine the number of smaller objects of known volume that will fit within a larger object of known volume, we must know the exact dimensions of both the smaller objects and the larger object.
→ Is it possible to determine the number of smaller boxes, each with a volume of 6cm3, that would fit within a larger box of 36cm3? No, boxes of different dimensions will fit into each other differently. Some of the combinations of dimensions may fit evenly into the large box, and some may fit into the large box but leave regions of empty space.
42
Sphere:
Surface Area and Volume
**Surface Area = 4πr2** (Eddie Woo's 4 circle, 3B1B's cylinder projection)
**Volume = (4/3) πr3**
(Spheres are not tested on GMAT)
43
Cone:
Surface Area and Volume
**Surface Area = πr2 + πrs** (s = slanted height)
**Volume = (⅓) πr2h** (h = perpendicular height; difficult proof)
(Cones are not tested on GMAT)
44
Pyramid:
Surface Area and Volume
**Surface Area = b + ps/2** (s = slanted height, p = perimeter of base)
**Volume = (⅓) bh** (h = perpendicular height, b = base area)
(Pyramids are not tested on GMAT)
45
Geometry Area: Hard Question
A square wooden plaque has a square brass inlay in the center, leaving a wooden strip of uniform width around the brass square. If the ratio of the brass area to the wooden area is 25 to 39, which of the following could be the width, in inches, of the wooden strip?
