Amber Kuang

Cheat Sheet / Updated 03-09-2022

Geometry is full of formulas, properties, and theorems. You can become successful in geometry by remembering the most important ones and learning how to apply them. Use this reference sheet as you practice various geometry problems to grow your knowledge and skills.

Article / Updated 03-26-2016

A polygon is a geometric figure that has at least three sides. The triangle is the most basic polygon. You will find the following formulas and properties useful when answering questions involving triangle inequalities, right triangles, relationships between the angles and sides of triangles, and interior and exterior angles of polygons. All triangles The sum of the three interior angles of a triangle is 180°. The largest side of a triangle is opposite the largest angle of the triangle. The sum of the two shorter sides of a triangle must be greater than the longest side of the triangle. The exterior angle of a triangle is equal to the sum of the two nonadjacent interior angles of the triangle. The centroid of a triangle divides each median of the triangle into segments with a 2:1 ratio. Right triangles The Pythagorean theorem states that a2 + b2 = c2, where a and b represent the legs of the right triangle and c represents the hypotenuse. When you draw an altitude to the hypotenuse of a right triangle, you form two right triangles that are similar to each other and also similar to the original right triangle. Because these triangles are similar, you can set up the following proportions: The altitude to the hypotenuse of a right triangle is the mean proportional between the two segments that the hypotenuse is divided into: The leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse: Here are the trigonometric ratios in a right triangle: Polygons The sum of the degree measure of the interior angles of a polygon equals 180(n – 2), where n represents the number of sides. The sum of the exterior angles of a polygon is 360°. The area of a regular polygon equals The apothem is the line segment from the center of the polygon to the midpoint of one of the sides.

Article / Updated 03-26-2016

Coordinate geometry is the study of geometric figures graphed on a coordinate plane. The slope formula can be used to determine whether lines are parallel or perpendicular. The midpoint can be used to determine if segments are bisected and also can be used to find the center of a circle. The distance formula can be used to determine the lengths of sides of geometric figures. Distance formula: Midpoint formula: Slope formula: Slope-intercept form of a line: Point-slope form of a line:

Article / Updated 03-26-2016

To solve geometry problems about circles, you will need to know the following circle theorems involving tangents, secants, and chords. These theorems can be used to find information about angles, intercepted arcs, and length of segments of a circle. In addition, you find the standard and general form of a circle, the formulas for area and circumference, and the area of a sector of a circle. Circle formulas The circumference of a circle equals The area of a circle equals The area of a sector equals Standard form of a circle: General form of a circle: Circle theorems involving angles The central angle equals the intercepted arc. An inscribed angle equals The interior vertical angles formed by two intersecting chords equal An exterior angle equals A line tangent to a circle is perpendicular to the radius drawn to the point of tangency. Circle theorems involving lengths of segments When a tangent and secant are drawn from the same exterior point, When two secants are drawn from the same exterior point,

Article / Updated 03-26-2016

Many formulas are associated with the study of three-dimensional shapes in geometry. Here, you find formulas for calculating the volume, surface area, and lateral area of cylinders, cones, spheres, pyramids, cube, and rectangular prisms. Cylinders The lateral area of a cylinder equals Cones The lateral area equals Spheres Square pyramids Cubes Rectangular prisms

Article / Updated 03-26-2016

In coordinate geometry problems, there are special rules for certain types of transformations. To determine the image point when performing reflections, rotations, translations and dilations, use the following rules: Reflections: Rotations: Translations: Dilations:

Article / Updated 03-26-2016

When two parallel lines are intersected by a third line (called a transversal), congruent pairs of angles are formed, including alternate interior angles and alternate exterior angles. The following practice questions ask you to use this information to find a missing angle, and then to apply some algebra to calculate a missing variable. Practice questions Use the diagram and the given information to solve the problem. Parallel lines are cut by a transversal. If is represented by is represented by 4x – 45, find the value of x. Parallel lines are cut by a transversal. If are represented by respectively, find the degree measure of Answers and explanations 20 are alternate interior angles. Alternate interior angles are congruent, so set their measures equal to each other and solve for x: 140 degrees are alternate exterior angles. Alternate exterior angles are congruent, so set their measures equal to each other and solve for x: which means they add up to 180 degrees. Use this info to solve for

Article / Updated 03-26-2016

Angles that form a linear pair combine to form a straight angle. (A straight angle measures 180 degrees.) The following practice questions ask you to solve problems based on linear pairs. Practice questions In the following figure, at E. In the following questions, fill in the blank to make the statement true. If you know that are represented by 2a, 2a + b, and 3a – 20, respectively then b = ———? If you know that Answers and explanations 20 Start with the given information: form a linear pair, which means their sum is 180 degrees. Set up the following equation and solve for a: Plug in the value of a to find because they’re vertical angles. Set them equal to each other, plug in the value of a, and solve for b: 124 degrees Angles that form a linear pair add up to 180 degrees. Set the sum of equal to 180 and solve for x: Now plug in the value of x to solve for

Article / Updated 03-26-2016

If you are asked to find the area of a regular polygon, you can do so by using a formula that includes the perimeter of the polygon and a measurement called the apothem. The apothem is the line segment from the center of the polygon to the midpoint of one of the sides, and is perpendicular to that side. The perimeter is the total distance around the polygon. The formula for the area of a regular polygon is Practice questions Find the area of a regular pentagon whose perimeter is 40 units and whose apothem is 5 units. Find the exact area of a regular hexagon that has a perimeter of 60 units. Answers and explanations 100 units2 The formula for the area of a regular polygon is The apothem is 5 and the perimeter is 40, so the area is The formula for the area of a regular polygon is A regular hexagon is a polygon with six equal sides. You're given that the perimeter of the hexagon is 60 units, which means each side is 10. The apothem is joined to the midpoint of one of the sides and is also perpendicular to the side, forming a The side opposite the 30-degree angle is x, the side opposite the 60-degree angle is and the side opposite the 90-degree angle is 2x. The apothem is opposite the 60-degree angle, so the apothem equals When you plug everything into the formula, you get

Article / Updated 03-26-2016

You can use your knowledge of geometric constructions (as well as a compass and straight edge) to create congruent angles. The following practice questions test your construction skills. If you're drawing two arcs for a construction, make sure you keep the width of the compass (or radii of the circles) consistent. Practice questions Use the above figure to construct an angle congruent to angle A. Use the above figure to construct a triangle congruent to triangle BCA. Answers and explanations 1.Here is the solution: Use a straight edge to draw a ray with endpoint D. Place the compass point on A, and with any width, draw an arc intersecting the angle at two points. Using the same width, place the compass point at D and make an arc. Using the compass, measure the distance between B and C. Keeping that compass width, place the compass point at E and draw an arc. Connect the point where the arcs intersect to D. 2.Here is the solution: Place the compass point at B and measure the length of Draw Point D on your paper. Keeping the length of place the compass point on D and draw an arc. Place a point on the arc and label it E. Use your compass to measure the length of Keeping that compass width, place the compass point at D and draw an arc where the third vertex would be located. Use your compass to measure the length of Keeping that compass width, place the compass point at E and draw an arc where the third vertex would be. Name the point where the arcs intersect F. Connect the three vertices of the triangle.