8. Geometry Flashcards
What is a Polygon?
- A two-dimensional, closed shape made of line segments
- Includes three-sided shapes (triangles), four-sided shapes (quadrilaterals) and other polygons with n sides (where n is five or more)
- Note that a circle is a closed shape but it is not a polygon because it does not contain line segments
What is the perimeter?
Sum of the lengths of all sides
What is a quadrilateral?
- Any figure with four sides
- Note that you can cut up any quadrilateral into two triangles by slicing them across the middle to connect opposite corners
What are special types of quadrilaterals?
- Trapezoids
- Parallelograms
- Special Parallelograms (Rhombuses, Rectangles, Squares)
*Note a square is both a rhombus and a rectangle
What is a parallelogram?
- Quadrilateral in which the opposite sides are parallel and equal
- Opposite angles are also equal and adjacent angles add up to 180
- Consists of a square and two identical right triangles
What is the area of a parallelogram?
Area = (base)(height)
*With parallelograms, as with triangles and trapezoids, remember that the base and the height must be perpendicular to one another
What is a rhombus?
- Quadrilateral in which all of the sides have the same length and in which the opposite angles are equal
- Every rhombus is a parallelogram, and a rhombus with right angles is a square
What is the area of a rhombus?
Area = (Diagonal1 * Diagonal2) / 2, where the diagonals refer to the lengths of the lines drawn between opposite vertices in the rhombus
*NOTE: the diagonals of a rhombus are always perpendicular bisectors (meaning they cut each other in half at a 90 degree angle
What is a rectangle?
- Have all the same properties of a parallelogram, plus one more – all four internal angles of a rectangle are right angles
- With rectangles, you refer to one pair of sides as the length and one pair of sides as the width
*Note: the diagonal of a rectangle cuts the rectangle into two equal right triangles, with all the properties you expect of right triangles
What are the perimeter and area of a rectangle?
Perimeter = 2(width + length) Area = width * length
What is a square?
- A rectangle in which all 4 sides are equal and all angles are 90 degrees
- Thus, knowing only one side of the square is enough to determine the perimeter and area of a square
*NOTE: Squares are like circles in that if you know one measure, you can find everything
What are the perimeter and area of a square?
Perimeter = 4(Side) Area = Side^2
What is a trapezoid?
- A quadrilateral with at least one pair of parallel sides
- The parallel sides are called bases and the other two sides are called the legs
*NOTE: a scalene trapezoid is a trapezoid with no sides of equal measure
What is the area of a trapezoid?
Area = h * (Base1 + Base2)/2
- Height refers to the line perpendicular to the two bases, which are parallel
- Note that you often have to draw in the height
What are the Interior Angles of a polygon?
- The angles that appear in the interior of a closed shape
- The sum of those angles depends only upon the number of sides in the closed shape:
- Sum of Interior Angles of Polygon = (n - 2) * 180, where n = the number of sides in the shape
What is the sum of the interior angles of a triangle?
3 sides, 180 degrees
What is the sum of the interior angles of a quadrilateral?
4 sides, 360 degrees
*NOTE: a quadrilateral can be cut into two triangles by a line connecting opposite corners
What is the sum of the interior angles of a pentagon?
5 sides, 540 degrees
*NOTE: a pentagon can be cut into three triangles by two lines connecting opposite corners
What is the sum of the interior angles of a hexagon?
6 sides, 720 degrees
*NOTE: a hexagon can be cut into four triangles by three lines connecting opposite corners
What does Two-dimensional mean?
A shape containing a length and a width.
What does Three-dimensional mean?
An object containing a length, a width, and a height.
What is a rectangular solid?
A three-dimensional shape consisting of six faces, at least two of which are rectangles (the other four may be rectangles or squares, depending upon the shape’s dimensions)
What is the Surface Area of a Rectangular Solid?
Surface area = the sum of the areas of all six faces
What is the Volume of a Rectangular Solid?
Volume = length * width * height, where length, width, and height refer to the three dimensions of the rectangular solid.
What is a Cube?
A three-dimensional shape consisting of six identical faces, all of which are squares
What is the Surface Area of a Cube?
Surface area = the area of any one face multiplied by 6.
NOTE: only need to know length of one side to determine surface area
What is the Volume of a Cube?
Volume = s^3, where s refers to the length of any one side of the cube
NOTE: only need to know length of one side to determine volume
How many books, each with a volume of 100 in^3, can be packed into a crate with a volume of 5,000 in^3?
BEWARE OF THE GMAT VOLUME TRICK. It is tempting to answer 50 books. However, this is incorrect, because you do not know the exact dimension of each book!
One book might be 554 while another book might be 2051 – both of which have a volume of 100 in^3. However, both have different rectangular shapes
Takeaway:
-When you are fitting 3-dimensional objects into other 3-dimensional objects, knowing the respective volumes is not enough – you must know the specific dimensions (length, width, height) of each object to determine whether the objects can fit without leaving gaps
Geometry Strategy Guide, Ch 1, Q 10. ABCD is a square picture frame. EFGH is a square inscribed within ABCD as space for a picture. The area of EFGH (for the picture) is equal to the area of the picture frame (the area of ABCD minus the area of EFGH). If AB = 6, what is the length of EF?
Area(EFGH) = Area(ABCD) – Area(EFGH) 2Area(EFGH) = Area(ABCD) 2Area(EFGH) = 6*6 Area(EFGH) = 18 = x^2 X = √18 = 3√2
What is a Triangle?
- A three-sided closed shape composed of straight lines
- The interior angles add up to 180°
How do you determine the area of a triangle?
Area of triangle = (1/2)(b)(h)
- Base refers to the bottom side of the triangle (note that ANY side of the triangle could act as a base)
- Height always refers to a line drawn from the opposite vertex to the base
- The base and the height must be perpendicular (form 90 degree angle) to each other
*The height can be outside the triangle! (You just have to extend the base, but not for the purposes of calculating the base in the area formula)
What is a Vertex (singular) or Vertices (plural)?
An “angle” or place where two lines of a shape meet; for example, a triangle has three vertices and a rectangle has four vertices
What are the Legs of a Triangle?
The smaller sides of a triangle; usually used in describing a right triangle, in which there is one hypotenuse (the longest side) and two legs (the shorter sides)
What is the Hypotenuse of a Triangle?
The longest side of a right triangle. The hypotenuse is opposite the right angle.
What is a Right Triangle?
- A triangle that includes a 90°, or right, angle
- Given the lengths of any two of the sides of a right triangle, you can determine the length of the third side using the Pythagorean Theorem
- With 30-60-90 and 45-45-90 right triangles, you only need the length of one side to determine the lengths of the other sides
*Right triangles are essential for solving problems involving other polygons
What are the two key properties of the angles of a triangle?
(1) Sum of the three angles of a triangle equals 180
(2) Angles correspond to their opposite sides
- Largest angle is opposite the longest side, while the smallest angle is opposite the shortest side and vice versa
- Additionally, if two sides are equal, their opposite angles are also equal (isosceles triangles)
What is the Triangle Inequality Law?
If you are given two sides of a triangle, the length of the third side must lie between the difference and the sum of the two given sides
- Any side of a triangle must be less than the sum of the other two sides
- Any side of a triangle must be greater than the difference of the other two sides
How do you determine the third angle of a triangle if you know two angles of a triangle or can represent all three in terms of x?
The sum of the internal angles of a triangle must add up to 180 degrees. As a result, if you know two angles of a triangle, you can find the third angle
Or you might be given one angle and the other angles in terms of x, in which case you solve the equation for x
What is a right triangle?
- Any triangle in which one of the angles is a right angle (90 degrees)
- Every right triangle is composed of two legs and one hypotenuse (side opposite the right angle, or c)
- APPLY PYTHAGOREAN’S THEOREM TO DETERMINE LENGTH OF SIDES
What is the Pythagorean Theorem?
a^2 + b^2 = c^2
-The lengths of the three sides of a right triangle are related by the equation above, where a and b are the lengths of the sides touching the right angle, also known as legs, and c is the length of the side opposite the right angle, also known as the hypotenuse
- Only applies to right triangles
- You can always find the length of the third side of a right triangle if you know the lengths of the other two sides
What are the Pythagorean triplets?
A subset of right triangles in which all three sides have lengths that are integer values a-b-c 3-4-5 or 6-8-10 or 9-12-15 or 12-16-20 5-12-13 or 10-24-26 7-24-25 8-15-17 9-40-41
Note: You can double, triple or otherwise apply a common multiplier to these lengths
What is an imposter triangle?
A non-right triangle with two sides equal to two parts of a Pythagorean triplet
Example: a triangle with one side equal to 3 and one side equal to 4 does NOT necessarily mean that the third side has a length of 5; this rule only applies to triangles that are KNOWN to be right triangles
What is an isosceles triangle?
- A triangle that has two equal angles and two equal sides (opposite the equal angles)
- An isosceles right triangle has one 90 degree angle (opposite to the hypotenuse) and two 45 degree angles (45-45-90 triangle)
What are the properties of a right isosceles triangle?
If the angles of a triangle are equal to 45, 45 and 90 degrees, then the lengths of the sides are proportional to x : x : x√2
Leg : Leg : Hypotenuse
X : X : X√2
*IMPORTANT: an isosceles triangle is equal to exactly one half of a square (i.e. two right isosceles triangles make up a square)
What is an equilateral triangle?
A triangle that has 3 angles (all 60 degrees) and 3 equal sides
What do we know about a triangle inscribed inside of a circle?
- If one length is equal to the diameter of the circle, then the triangle is a right triangle
- The angle opposite to the hypotenuse is equal to 90 degrees
- Otherwise the triangle is not a right triangle
What are the properties of a 30-60-90 triangle?
If the angles of a triangle are equal to 30, 60 and 90 degrees, then the lengths of the sides are proportional to x : x√3 : 2x
Short Leg : Long Leg : Hypotenuse
X : X√3 : 2X
Given that an equilateral triangle has a side length of 10, what is its height?
- The side of an equilateral triangle is the same as the hypotenuse of a 30-60-90 triangle
- The height of an equilateral triangle is the same as the long leg of a 30-60-90 triangle
- Thus, the long leg has a length of 5√3, which is the height of the equilateral triangle
What is the area of an equilateral triangle?
In addition to the standard area formula for triangles, equilateral triangles have a special formula for area:
Area = (√3 * S^2)/4, where S is the length of any side of the equilateral triangle.
What do you do if you see two equal sides in a triangle?
Set the opposite angles equal to each other
What do you do if you see two equal angles in a triangle?
Set the opposite sides equal
What is the Diagonal of a Square?
d = s√2, where s is a side of the square
*This can also be the face diagonal of a cube
What is the Main Diagonal of a Cube?
- The main diagonal of a cube is the one that cuts through the center of the cube
- The diagonal of a face of a cube is not the main diagonal
d = s√3, where s is an edge of the square
What is the measure of an edge of a cube with main diagonal of length √60?
√60 = s√3 s = √60 / √3 = √20
What is the diagonal of a rectangle?
To find the diagonal of a rectangle, you must know either the length and width or one dimension and the proportion of one to the other
Using Pythagorean’s theorem, c = √(a^2 + b^2)
What is the Main Diagonal of a Rectangular Solid?
- The main diagonal of a rectangular solid is the one that cuts through the center of the solid
- The diagonal of a face of the rectangular solid is not the main diagonal
- The main diagonal of a rectangular solid can be found by using the “Deluxe” Pythagorean Theorem: d^2 = x^2 + y^2 + z^2, where x, y, and z are the length, width, and height of the rectangular solid, and d is the main diagonal
What are Similar Triangles?
-Triangles in which the three corresponding angles are identical and the corresponding sides are in proportion
What are the properties of Similar Triangles?
(1) 2 Angles: It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180 degrees
(2) If two similar triangles have corresponding side lengths (or heights or perimeters) in ratio a:b, then their areas will be in ratio a^2:b^2
What is the principle regarding Similar Figures?
For similar solids (e.g. quadrilaterals, pentagons, etc) with corresponding sides in ratio a:b, their volumes will be in ratio a^3:b^3
How do you determine the length of the third side of a right triangle if you know the other two sides?
(1) Can find the third side, either by using the full Pythagorean theorem or by recognizing a triplet
(2) Make that unknown side less than the sum of the other two sides but more than the difference
Geometry Strategy Guide, Ch 2, Q 3. Triangle A has a base of x and height of 2x. Triangle B is similar to Triangle A, and has a base of 2x. What is the ratio of the area of Triangle A to Triangle B?
1 to 4. You can use the shortcut or calculate the Area for each triangle in terms of x.
Ratio of Areas = a^2 + b^2 = 1^2 : 2^2 = 1 : 4
Area(A) = (1/2)(x)(2x) = x^2
Area(B) = (1/2)(2x)(4x) = 4x^2
Ratio of Areas = 1:4
Geometry Strategy Guide, Ch 2, Q 6. In Triangle ABC, AD = DB = DC. Given that angle DCB is 60 degrees and angle ACD is 20 degrees, what is the measure of angle x?
If AD = DB = DC, then the three triangular regions in this figure are all isosceles triangles. Therefore, you can fill in some of the missing angle measurements. Since you know that there are 180 degrees in the large triangle ABC, you can write the following equation:
x + x + 20 + 20 + 60 + 60 = 180
2x + 160 = 180
x = 10
Geometry Strategy Guide, Ch 2, Q 8. What is the area of an equilateral triangle whose sides measures 8cm long?
Area Equilateral Triangle = (√3 * S^2)/4 = 16√3