Geometry Flashcards
Right Triangles:
Right triangles (Rechtwinkliges Dreieck) are frequently tested on the GMAT. In a right triangle one angle is 90 degrees.
Pythagorean theorem to calculate the sides of a right triangle CDE where CD is the hypotenuse (longest side of a right-angled triangle):
(ED)^2 + (CE)^2 = (CD)^2 OR in triangle abc: a^2 + b^2 = c^2
A 30-60-90 triangle is a special kind of triangle in which the hypotenuse is twice the length of one side of the triangle. Here the lengths of the triangle will be in the ratio 1:√3:2 where 1 is one side, √3 the other and 2 the hypotenuse.
Lines:
The MIDPOINT is the point that divides a line segment into two equal parts.
A------------M-------------B
AB = AM + BM
Two PARALLEL Lines will never meet. Write:
L1 II L2
Two lines are PERPENDICULAR (lotrecht) if they intersect in a 90 degree angle. Write:
L1 I L2.
If L1 I L2 and L2 I L3, then L1 II L3.
Angles (Winkel):
Two intersecting lines form an angle and the point of intersection is called vertex (Scheitel) of the angle. Angles are measured in degrees.
Acute Angle: degree of angle is between 0° and 90°
Right Angle: degree of angle is exactly 90°
Obtuse Angle: degree of angle is between 90° and 180°
Straight Angle: degree of angle is exactly 180° (half a circle)
Sum of Angle Measures:
- Sum of measures of angles on one side of a straight line is 180°
z x \ / z \_\_\_\/\_\_\_ x + y + z = 180 - Sum of all the angles around a point is 360°
c
b \ / d
\_\_\_\/\_\_\_ a + b + c + d + e = 360°
a /\ e
/ \
f
- Two angles are supplementary if together they make up a straight angle (i.e. sum of angles measure 180°)
/ / c / d c + d = 180° supplementary -------------------/-------------------- - Two angles are complementary if together they make up a right angle (i.e. sum of angles measure 90°)
I a / I / I / b a + b = 90° complementary I/------------------ - A line bisects an angle if it splits the angle into two smaller, equal angles. ABD has same measure of DBC. ABD and DBC are each half of the size of ABC.
A \ / D \ / \/-------------C B
Vertical Angles:
- Vertical angles are a pair of opposite angles formed by two intersecting lines. Pairs a and c are vertical angles, so are b and d.
Vertical angles are equal.\ b / \ / a /\ c a + b = c + d = a + d = b + c = 180° / d \
Each angle is supplementary to each of its adjacent angles, i.e. together the two angles make up 180°. (e.g. a + b = 180°)
- If two parallel lines intersect with a third line (called transversal) the corresponding angles are equal:
a = e, c = a, e = g (vertical angles) therefore: a = c = e = g
and b = d = f = h
b / a
------------------------/----------------
c / d
/
/
f / e
----------------/------------------------
g / h
I.e. when two parallel lines intersect with a third line, all acute angles formed are equal, all obtuse angles formed are equal, any acute angle is supplementary to any obtuse angle. When a transversal cuts through a pair of parallel lines, knowing the measures of one of the eight angles formed allows us to determine the measure of any of them, since all acute angles are equal and all obtuse angles are equal.
Triangles:
The sum of interior angles of any triangle is 180°. Each interior angle is supplementary to an adjacent exterior angle. The degree measure of an exterior angle is equal to the sum of the measures of the two nonadjacent (remote) interior angles.
a a + b + c = 180°
/ \ d is supplementary to c
/ \ d + c = 180°
/ c \ d d + c = a + b + c
b /-----------\-------- d = a + b
REMEMBER: Angles correspond to their opposite sides. The largest angle is opposite the largest side, the smallest angle opposite the smallest side. If two sides are equal their opposite angles are also equal. Mark equal sides and angles with a slash in when you redraw the figure on scrap paper.
REMEMBER: look for exterior angles in complicated figures. Remove lines and draw others to find relationships between angles.
Altitude of Triangle:
The altitude or height of a triangle is the perpendicular distance from a vertex (Scheitel) to the side opposite the vertex, also das Lot vom Scheitel zur gegenueberliegenden Seite des Dreiecks. The altitude can fall inside, outside or on one of the sides of the triangle.
A
/\
E.g. / I \
/ I \
B /—–I——\ C Altitude = AD
Sides of Triangles:
B b + c > a > | c - b |
/ \
a / \ c
/ \
C /———–\ A
b
If the lengths of two sides of a triangle are unequal, the greater angle lies opposite the longer side. if C > A > B, then c > a > b.
REMEMBER: The sum of any two sides in a triangle must ALWAYS be greater than the third side. Because i.e. in figure above, the shortest way between A and B is AB. Making the detour of going to C first and then to A must mean that BC + CA must be greater than BA. This also means that the sum of the two smaller sides in a triangle is greater than the largest side of the triangle.
This also means: The length of one side cannot be shorter than a certain length, in particular it has to be longer than the difference between the two other sides. In figure above, c > a-b and b>a-c and a>c-b (what is subtracted from what becomes clear when you have actual values for the sides in particular triangle).
THAT MEANS: 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.
Area of Triangle:
Formula:
1/2 x base x height.
The base is the bottom side of triangle. The height ALWAYS refers to a line drawn from the opposite vertex (Scheitel) to the base, creating a 90° angle.
/\
E.g. / I \
/ I 3 \
/----I------\ Area = 1/2 x 4 x 3 = 6
4
REMEMBER: You can designate any side of the triangle as the base. That means that all triangles have 3 possible bases. And depending on which side you pick to be the base the height can either fall within or outside of the triangle. Remember height is always perpendicular from base to opposite vertex (Scheitel).
When two sides of a triangle are perpendicular to each other the area is easy to find. E.g. in a right triangle look at the two sides that form then 90° angle (called legs) and the area is 1/2 the product of the legs.
Area Formula for Right Triangles:
1/2 x L1 x L2 L = Legs of the two sides forming the 90° angle. And it doesn't matter which side you consider the base and which the height.
Perimeter of Triangle:
The perimeter is equal to the sum of the lengths of the 3 sides.
Isosceles (gleichschenklig) Triangle:
An isosceles triangle is a triangle that has two sides of equal length. The two equal sides are the LEGS and the third side is the BASE.
P
/\ PQ = PR and Q = R
/ \
/ \
Q /----------\ R
Since the two legs have the same length the two angles opposite the legs must have the same measure.
Equilateral (gleichseitiges) Triangles:
Equilateral triangles have three sides of equal length and three 60° angles.
Congruent (deckungsgleich) Triangles:
Triangles are congruent if corresponding angles have the same measure and corresponding sides have the same length.
Similar Triangles:
Triangles are similar if all corresponding angles have the same measure. REMEMBER: Once you find that two triangles have two equal angles, you know that the third angle is also equal between the too.
Plus, in similar triangles, corresponding side lengths are proportional to one another. E.g. if in one triangle the base has the length of 6 and you know that the corresponding side in a similar triangle (so the base of that similar triangle) is twice as long then you know that all sides of that other triangle are twice as long as the corresponding sides in the first triangle.
similar triangles are a very helpful tool for GMAT so learn to identify them!!!
The ratio of the areas of two similar triangles is the square of the ratio of any of the corresponding lengths. E.g. in two similar triangles where in one (ABC) the corresponding sides are twice as long as in the other one (DEF):
Area ABC/Area DEF = (DE/AB)^2 = (2/1)^2 = 4
DE and AB are two corresponding sides in the respective triangles.
That means ABC has 4 times the area of DEF.
REMEMBER: in similar triangles, because they have proportional side lengths) you can find the length of sides by setting sides in proportion to each other and using info on side lengths given in stem to find new side lengths (e.g. AB/AC = AE/AC)
Geometry Extra: Corresponding Areas in Similar Triangles:
General Rule:
If two similar triangles have corresponding sides a and b in ratio a:b, then their areas will be in ratio a^2:b^2
E.g.
Two legs of right triangle 6: 12 and 9
Two legs of right triangle 2: 4 and 3
You can see instantly that these are similar triangles with corresponding sides in ratio 3:1 (for triangle1:triangle2)
If you calculate the areas for the triangles you get:
Area of triangle 1: 54
Area of triangle 2: 6
The area of triangles is in ratio 54:6 = 9:1
So, you can see while the lengths are in ratio 3:1, the areas are in ratio 3^2:1^2 = 9:1
This rule also holds when you take the height or perimeters. The rule also holds true for similar polygons, quadrilaterals, pentagons etc.
REMEMBER: in similar triangles, because they have proportional side lengths) you can find the length of sides by setting sides in proportion to each other and using info on side lengths given in stem to find new side lengths (e.g. AB/AC = AE/AC)
Right Triangles:
Right triangles (Rechtwinkliges Dreieck) are frequently tested on the GMAT. In a right triangle one angle is 90 degrees which is also the largest angle of the triangle
Pythagorean theorem to calculate the sides of a right triangle ABC where AC is the hypotenuse (longest side of a right-angled triangle) and ABC is where the 90° is:
(AB)^2 + (BC)^2 = (AC)^2 or (Leg1)^2 + (Leg2)^2 = Hypotenuse^2
Example of how to use the Pythagorean theorem on GMAT:
What is the length of the hypotenuse of a right triangle with legs of length 9 and 10?
Hypothenuse^2 = 9^2 + 10^2
Hypothenuse^2 = 81 + 100
Hypothenuse^2 = 181
I.e. Hypothenuse = √181
Common Right Angles on GMAT:
All these satisfy the Pythagorean theorem and are often testes on GMAT so memorize:
- Triangle with sides: 3, 4, and 5, where 3 and 4 are the legs and 5 is the hypotenuse (most common kind of right triangle tested on the GMAT)
3^2 + 4^2 = 5^2
9 + 16 = 25
Also, some multiples of these lengths makes a Pythagorean triple.
E.g. 6^2 + 8^2 = 10^2
36 + 64 = 100
So 6, 8, and 10 also make a right triangle.
or Multiples of 6, 8, 10:
9^2 + 12^2 = 15^2 81 + 144 = 225 12^2 + 16^2 = 20^2 144 + 256 = 400
- Triangle with sides 5, 12, 13
5^2 + 12^2 = 13^2
25 + 144 = 169
or Multiple of 5, 12, 13 like 10, 24, 26
- Triangle with sides 8, 15 and 178^2 + 15^2 = 17^2
64 + 225 = 289
NO Multiples of this one satisfy the Pythagorean theorem.
Special Right Triangles:
You can always use the Pythagorean theorem to find the lengths of the sides in a right triangle. But there are two special kinds of right triangles that always have the side lengths in the same ratio:
- A 30-60-90 triangle is a special kind of triangle in which the hypotenuse is twice the length of one side of the triangle. Here the lengths of the triangle will be in the ratio 1:√3:2 where 1 is one side, √3 the other and 2 the hypotenuse. Or any multiple of this ratio.
Proportions of side lengths of 30-60-90 triangle: x:x√3:2x
- In isosceles right triangles, also known as 45-45-90 triangles have the lengths in the ratio 1:1:√2 where √2 is the length of the hypotenuse. Or any multiple of this ratio.
Proportions of side lengths of 45-45-90 triangle: x:x:x√2
But GMAT can hide these two in other shapes to trick test takers.
a) two isosceles (gleichschenklig) right triangles put together form a square and S√2 is the diagonal of the square (Where S is a side of the square). b) An equilateral (gleichseitig) triangle divided into two equal parts creates two 30, 60, 90 triangles.
IMPORTANT: The GMAT tests special right angles all the time. Watch out for them and use critical thinking and pattern recognition to find them on the test and save a lot of time and calculation.
GMAT tests these all the time:
Right triangle with side length: 3:4:5 (or multiple, like 6:8:10)
Right triangle with side length 5:12:13
Isosceles right triangle (45-45-90) with side length ratio: 1:1:√2 (√2 is hypo)
30-60-90 right triangle with side length ratio: 1:√3:2 (2 is hypo)
Geometry Extra: Area of Equilateral Triangle:
An equilateral (gleichseitig) triangle (three equal sides and three 60° angles) divided into two equal parts creates two 30-60-90 triangles. Two sides (S) of the equilateral then form the hypothenuse of each of the 30-60-90 triangles. And the second longest leg of each triangle form the height of the equilateral triangle. You know that a 30-60-90 triangle has the side ratios x:x√3:2x. So in the equilateral triangle with side S the two 30-60-90 triangles have the side lengths:
1/2S (since the base side S is divided by 2 through the two
triangles)
1/2S√3 (longer leg)
2S (hypothenuse)
So, the Area of the equilateral triangle with side length S is:
Area = 1/2 x base x height
Area = 1/2 (S) (1/2S√3) = 1/4 S^2 √3 = S^2√3/4
Polygons:
Number of sides determines the name of the polygon:
3 sides: Triangle (sides don’t have to be same size)
4 sides: Quadrilateral (sides don’t have to be same size)
5 sides: Pentagon (sides don’t have to be same size)
6 sides: Hexagon (sides don’t have to be same size)
Triangles and quadrilaterals are by far most tested on GMAT.
In a regular polygon all sides and all vertices have the same measurements.
Formula to find sum of interior angles of polygons with n sides:
(n-2)180°
The measure of each interior angle then is:
(n-2)180° ÷ n
In a quadrilaterial, polygon with 4 sides, the angles sum up to 360°. (that’s also because quadrilaterals can be cut into two equal triangles which means the angles have to sum up to 2x180° = 360°)
In a Pentagon, polygon with 5 sides, the angles sum up to 540°.
In a Hexagon, polygon with 6 sides, the angles sum up to 720°. (that’s also because hexagons can be cut into 4 equal triangles by three lines which means the angles have to sum up to 4x180° = 720°)
Non-Convex Polygons:
REMEMBER: The sum of the exterior angles for any non-convex polygon is always 360.
Quadrilateral:
Sum of interior angles is 360°.
Most important quadrilaterals on GMAT: rectangles and squares.
RECTANGLE: all angles are 90°, opposite sides have the same length.
Square: Rectangle with 4 equal sides.
Area of Rectangle and Square:
Area of rectangle = length x width
Area of square = (side)^2
Parallelogram:
A parallelogram is a quadrilateral whose opposite sides are parallel. Parallelograms include rectangles and squares.
In a parallelogram the opposite sides are equal and opposite angles are equal. Adjacent angles add up to 180°
Area Formula for Parallelogram:
Area of Parallelogram = base x height
where the base is the bottom side and the height is the perpendicular to the base forming a 90° angle.
The diagonals of parallelograms are not necessarily the same length (they are in rectangles and squares) but they do bisect (in haelfte schneiden) each other.
