Geometry GMAT Concepts

Geometry concepts for GMAT include:

Lines and Angles

• A line is a one-dimensional abstraction that goes on forever.
• For any two points, there is one straight line (only one!) that passes through them.
• A line section is a segment of a straight line that has two endpoints. The midpoint is the point that divides the line segment into two equal parts.
• Two lines are parallel if they lie in the same plane and never intersect. Two lines are perpendicular if they intersect at a 90° angle.

• An angle is made when two lines intersect at a point. This point is called the vertex of the angle.
• Angles are measured in degrees (°).
• An acute angle is an angle whose degree measure is less than 90°.
• A right angle’s degree measure is exactly 90°.
• An obtuse angle’s degree measures greater than 90°.
• A straight angle’s degree measure is 180°.

• The sum of the measures of angles on a straight line is 180°.
• The sum of the measures of the angles around a point (which make a circle) is 360°.
• Two angles are supplementary if their sums make a straight angle.
• Two angles are complementary if their sums make a right angle.

• Vertical angles are opposite angles formed by two intersecting line segments.
• A line or a segment bisects an angle if it splits the angle into two, smaller equal angles.
• Vertical angles are a pair of opposite angles formed by intersecting line angles. The two angles in a pair of vertical angles have the same degree measure.

Triangles

• A triangle is a closed figure with three angles and three straight sides.
• The sum of the interior angles of a triangle is 180°.
• Each interior angle is supplementary to an adjacent exterior angle. Together, they equal 180°.
• The formula for finding the area of a triangle is ½bh.
  • b = base
  • h = height

• An isosceles triangle has two sides of equal length.

• An equilateral triangle has three equal sides and three angles of 60°.

• There are two kinds of special right triangles:
  • Isosceles right triangles have a side relationship of 1:1:√2.
  • 30°60°90° triangles have a side relationship of 1:√3:2.
• A right triangle has one 90° interior angle. The side opposite the right angle is called the hypotenuse and it’s the longest side of the triangle.
• Pythagorean Theorem for finding side lengths of a right triangle: a2+b2=c2
• Two triangles are similar if their corresponding angles have the same degree measure.
• Two triangles are congruent if corresponding angles have the same measure and corresponding sides have the same length.

Circles

• The diameter of a circle is a line segment that connects two points on the circle and passes through the center of the circle.
• The radius is a line segment from the center of the circle to any point on it.
• A circle’s central angle is formed by two radii.
• The distance around the circle is called circumference:
  • C=πd
  • C=2πr

• An arc is a part of the circumference of a circle.
  • Length=(n/360 °)C, where n is the measurement of the central angle of the circle portion in degrees.

• The area of a circle is found with the formula A=πr2.

Polygons

• A polygon is a closed figure that has straight line segments as its sides.
• The perimeter of a polygon is the distance around the polygon (the sum of the length of all its sides).
• The sum of the four interior angles of a quadrilateral is 360°.
• Area of a square: s2
• Area of a rectangle: lw
• Area of a parallelogram: bh

• Area of a trapezoid: 1/2(a+b)h

Solids

• A cylinder is a solid whose horizontal cross section is a circle.
• Volume of a cylinder: Bh, where B is the area of the base.
• Area of the base of a cylinder: ?r2 (because, remember, a cylinder has a circular cross section)
• A cube is a rectangular solid where all the faces are squares.
  • Volume of a cube: Bh, where B is the area of the base.

• A rectangular solid is a solid with six rectangular faces.
  • Volume of a rectangular solid: lwh

Coordinate Geometry

• GMAT Coordinate Geometry is the largest part of the field needed in GMAT.
• The slope of a line tells you how steeply that line goes up or down the coordinate plane.
  • slope = rise/run
  • slope=change in y/change in x$
• The rise is the difference between the y-coordinate values of two points on the line; the run is the difference between the x-coordinate values of two points on the line.
• You can also find the slope of a line using the slope-intercept equation, which is y=mx+b, where the slope is m and b is the value of the y-intercept.
• Perpendicular lines have slopes that are negative reciprocals of one another.
• To determine the distance between any two points on a coordinate plane, you can use the Pythagorean theorem.