TTP Geometry

Cutting an equilateral triangle in half forms two 30-60-90 triangles

Dropping an altitude from the upper vertex to the base of an equilateral triangle produces 2 identical 30-60-90 triangles.

The exterior angles of a triangle

Exterior angles of a triangle: angle created by one side of a triangle and the extension of an adjacent side. Sum of exterior angles of one triangle equals 360 degrees (works on all polygons). But each exterior angle must be less than 180. Only one angle per vertex for this to work.

Pythagorean Theorem

Hypotenuse: the side opposite the right angle; always the longest side. Other, shorter legs are called the legs. Length of hypotenuse squared (c²) = sum of other two sides squared (a² + b²).

The triangle inequality theorem

IN any triangle, the sum of the lengths of any two sides of the triangle is greater than the length of the third side, and the difference of lengths of any 2 sides of the triangle is less than the length of the third side.

Squares

Square: 4 equal sides and four equal angles, each of 90 degrees. Has all properties of parallelograms and rectangles. The diagonals of a square are perpendicular to each other and bisect each other. Each diagonal divides the square into two 45-45-90 right triangles. Diagonal of square: d=s√2 (s=length of a side)

Circles inscribed in squares

When a circle is inscribed in a square, a diameter of the circle has the same length as a side of the square.

Rectangles inscribed in circles

When a rectangle is inscribed in a circle, a diagonal of the rectangle is also a diameter of the circle. Unless the ratio of the shorter side of the rectangle to the longer side is x:x√3, they aren't 30-60-90 right triangles.

Interior angles of a triangle

3 interior angles of a triangle sum to 180 degrees

A regular hexagon can be divided into six equilateral triangles

A regular hexagon can be divided into six equilateral triangles.

Trapezoids

A trapezoid is a quadrilateral in which one pair of opposite sides are parallel but the other pair of opposite sides are not parallel. If 2 nonparallel sides are equal in length, the trapezoid is referred to as an isosceles trapezoid.

The equilateral triangle

All the angles are 60 degrees, and all of the sides are the same length. To find the area of an equilateral triangle, use the formula A=(s² x √3)/4, where s is the length of one side of the triangle.

Calculating the height of a triangle using an altitude of the triangle

Altitude is a synonym for height. For finding area, the base of the triangle must be perpendicular to the height. To determine the height, we must sometimes look to lines outside of the triangle itself.

Supplementary angles

Angles are supplementary if their measures add up to 180 degrees.

Area of a square

Area of a square = side²

The area of a triangle

Area of a triangle: 1/2(base x height) or 1/2ba. The base of the triangle is always perpendicular to the height.

Finding the area of a parallelogram

Area of parallelogram: b x h; the height is always perpendicular to the base

Three equivalent circle ratios

Central angle: any angle at the center of the circle that is formed by two radii Arc: portion of the circumference of a circle Sector: region of a circle defined by two radii and their intercepted arc central angle/ 360 = arc length/ circumference = area of sector/ area of circle

Circles

Circumference of circle: 2πr or πd Area of a circle: πr²

Polygons

Closed, 2-D geometric shape that is composed solely of straight line segments (triangles, squares, rectangles, etc.)

Similar triangles

If one triangle is simply and enlargement of another triangle, the triangles are similar. Similar if: 1) The 3 angles of one triangle are the same measure as the three angles of another triangle 2) the 3 pairs of corresponding sides have lengths in the same ratio 3) 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

Arcs

If points A and B are 2 points on a circle and arc AB is not a semicircle (doesn't cut the circle in half), arc AB refers to the shorter portion of the circumference between A and B. This shorter portion is the minor arc. Longer portion is the major arc, but usually referred to as arc ACB, C representing some third point.

Shaded regions

In general, when a geometrical figure has both shaded and unshaded regions, (the area of the entire figure) - (the area of the unshaded region) = (the area of the shaded region)

The isosceles right triangle (45-45-90 right triangle)

Isosceles: two equal sides and two equal angles Area of isosceles triangle: A= l²/2, where l is the length of one of the equal sides.

Triangles inscribed in a square

One side of triangle will coincide with one side of the square. Then the side of the triangle and the side of the square are of equal lengths, and the vertex of the triangle opposite the coincident sides touches the opposite side of the square.

The parallelogram

Parallelogram: a quadrilateral that has 2 pairs of parallel sides. 1) Opposite sides are equal in length 2) Opposite angles are equal in measure 3) Diagonals bisect each other 4) Each diagonal divides the parallelogram into two congruent triangles 5) Any 2 consecutive angles within a parallelogram are supplementary

Perimeter of a square

Perimeter of a square = 4 x side

Quadrilaterals

Quadrilaterals: 4-sided polygons -- rectangles, squares, parallelograms, rhombuses, and trapezoids. All squares are rectangles, and all rectangles are parallelograms.

Rectangle

Rectangle: any quadrilateral with four right angles Area of rectange: length x width Perimeter of rectangle: sum of all four sides, or P= 2l x 2w

Perpendicular lines

The angles formed by the intersection of perpendicular lines are 90 degrees.

Area of a regular hexagon

The area of a regular hexagon: ((3√3)/2)s², where s is the length of any of the hexagon's sides.

Area of a trapezoid

The area of a trapezoid: (base1 + base2)height/2. Remember that the bases of a trapezoid are parallel.

Inscribed angles

The degree measure of an inscribed angle is equal to half of the degree measure of the arc that it intercepts.

The area of a 45-45-90 right triangle is one-half of the area of a square

The diagonals of a square cut the square into two 45-45-90 right triangles. The are of each of those triangles is half of the are of the square that they form.

Relationship between angles and sides within a triangle

The largest angle is always opposite the longest side of the triangle; the smallest angle is opposite the shortest side of the triangle. In addition, equal sides will always be opposite equal angles.

The 30-60-90 right triangle

The lengths of the sides of a 30-60-90 triangle are in a ratio x:x√3:2x, where x represents the length of the side opposite the 30° angle, x√3 represents the length of the side opposite the 60° angle, and 2x represents the length of the side opposite the 90° angle.

The longest line segment of a rectangle

The longest line segment that can be drawn within a rectangle is its diagonal. The length of the diagonal of any rectangle is equal to the square root of (l² + w²). Each diagonal is the same length, and each diagonal bisects the other. In addition, each diagonal divides the rectangle into two congruent right triangles, which have the same area.

The exterior angles of any polygon sum to 360°

The measures of the exterior angles of any polygon, when taking one exterior angle at each vertex, always add up to 360°.

The ratio of the sides of a 45-45-90 right triangle

The sides of a 45-45-90 right triangle are in a set ratio of x:x:x√2, where x√2 represents the length of the hypotenuse and x represents the length of each of the shorter legs.

The interior angles of a polygon

The sum of the interior angles of a polygon = (n-2)x180, where n equals the number of sides in the polygon. The measure of any one interior angle in a regular polygon = 180(n-2)/n, where n is the number of sides in the polygon.

Hexagons

The sum of the interior angles of any hexagon is 720 degrees. Any one interior angle of a regular (all sides equal) hexagon measures 120 degrees.

Surface area of a rectangular solid or cube

The surface area of a cube = 6s², where s is the length of one side (or edge) of the cube. The surface area of a rectangular solid = 2(lw) + 2(lh) + 2(hw).

The right circular cylinder

The type of cylinder on the GRE. 2 circular bases are perpendicular to the height.

Volume and surface area of a right circular cylinder

The volume of a right circular cylinder = πr²h. The surface area of a right circular cylinder = 2(πr²) +2(πrh).

Common volume-rate traps

To determine the rate at which a liquid will rise within a 3D object, we must know the rate at which the liquid flows into the figure and the exact dimensions of the figure (total volume won't help here--a tall and skinny or short and fat with same volume won't have higher water level at same rate). 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.

The longest line segment that can be drawn within a rectangular solid or cube

To find the longest line segment that can be drawn within a rectangular solid, use the extended Pythagorean theorem. d²= l² + w² + h² When the solid is a cube, use d=s√3, where s is the length of one side (edge) of the cube.

Parallel lines intersected by a transversal

Transversal = a line that passes through 2 or more lines at different points 1)Vertical angles are equal 2) Corresponding angles are equal 3) Supplementary angles sum to 180 degrees 4) Any acute angle and any obtuse angle will sum to 180 degrees

Supplementary angles add up to 180 degrees

Two angles that are supplementary must add up to 180 degrees

Vertical angles are equal and corresponding angles are equal

Vertical angles: angles that are diagonally oriented to each other Corresponding angles: 2 non-adjacent angles that are on the same side of the transversal, but one is inside the parallel lines while the other is outside the parallel lines.

Volume of a cube or rectangular solid

Volume of cube = (edge)³²Volume of rectangular solid = length x width x height

Regular polygons inscribed in circles

When a regular polygon is inscribed in a circle, the polygon divides the circle into arcs of equal length.

Squares inscribed in circles

When a square is inscribed in a circle, a diagonal of the square is also a diameter of the circle.

Triangles inscribed in a circle

When a triangle is 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 90° angle opposite the diameter.

Intersecting lines

When n lines intersect through a common point, the sum of all the angles created by those n lines at that point is 360 degrees.

The 3-4-5 right triangle

When the sides of a triangle are in the ratio of 3:4:5, it is a 3-4-5 right triangle. Thus, triangles with the following side lengths are considered to be 3-4-5 right triangles: (3,4,5), (6,8,10), (9,12,15), etc.

The 5-12-13 right triangle

When the sides of a triangle are in the ratio of 5:12:13, it is a 5-12-13 right triangle. Thus, triangles with the following side lengths are considered to be 5-12-13 right triangles: (5,12,13), (10,24,26), (15,36,39), etc.

The cube and the rectangular solid

cube = 3D shape with 6 square sides (faces) rectangular solid = 3D shape with 6 rectangular faces in which opposite faces are of equal areas

Volume and rate

time = volume of container/ rate