The GRE General Exam - The Definitive Shortcuts Guide ALGEBRA is conceived as the second chapter of the three we will release every Monday in the coming weeks. It is not only just a collection of simple tricks or shortcuts for the quantitative reasoning portion of the GRE exam, but also a compendium if the students want to have, for example, the divisibility rule at a glance without consulting our GRE Math Essentials - A most comprehensive handout!! [COMPLETED] or the GRE - Math Book which I strongly recommend studying for complete and robust quant theory knowledge for the GRE.

The subsequent chapters that will be released are:

• NUMBER SYSTEM Shortcuts
• Algebra Shortcuts

(i) Sum of all the exterior angles of a polygon \(= 360°\)

(ii) Each exterior angle of a regular polygon \(= \frac{360°}{n}
\)

(iii) Sum of all the interior angles of a polygon \(= (n – 2) × 180°\)

(iv) Each interior angle of a regular polygon \(=\frac{ (n – 2)}{n} × 180°\)

(v) No. of diagonals of a polygon \(= \frac{n(n-3)}{2} \), n \(\rightarrow \)no. of sides.

(vi) The ratio of the sides of a polygon to the diagonals of a polygon is \(2 : (n – 3)\)

(vii) Ratio of the interior angle to the exterior angle of a regular polygon is \((n – 2) : 2\)




Properties of triangle:

(i) When one side is extended in any direction, an angle is formed with another side. This is called the exterior angle. There are six exterior angles of a triangle.

(ii) Interior angle + corresponding exterior angle = 180°.

(iii) An exterior angle = Sum of the other two interior opposite angles.

(iv) Sum of the lengths of any two sides is greater than the length of the third side.

(v) The difference between any two sides is less than the third. The side opposite to the greatest angle is the greatest and vice versa.

(vi) A triangle must have at least two acute angles.

(vii) Triangles on equal bases and between the same parallels have equal areas.

(viii) If a, b, c denote the sides of a triangle then

(i) if \(c^2< a^2 + b^2\) , Triangle is acute angled.

(ii) if \(c^2= a^ + b^2\) , Triangle is right angled.

(iii) if \(c^2> a^2+ b^2\), Triangle is obtuse angled.

If 2 triangles are equiangular, their corresponding sides are proportional. In triangles ABC and XYZ, if \(\angle A = \angle X, \angle B = \angle Y, \angle C = \angle Z\), then

\(\frac{AB}{XY}=\frac{AC}{XZ}=\frac{BC}{YZ}\)

(i) In \(\triangle ABC\), \(\angle B = 90° BD \perp AC \)

\(\therefore BD × AC = AB × BC\)




(ii) \(\frac{1}{BD^2}=\frac{1}{AB^2}+\frac{1}{BC^2}\)


(iii) \(BD^2=AD \times DC\)

(x) The perpendiculars drawn from vertices to opposite sides (called altitudes) meet at a point called the Orthocentre of the triangle.

(xi) The line drawn from a vertex of a triangle to the opposite side such that it bisects the side is called the Median of the triangle. A median bisects the area of the triangle.

(xii) When a vertex of a triangle is joined to the midpoint of the opposite side, we get a median. The point of intersection of the medians is called the Centroid of the triangle. The centroid divides any median in the ratio 2 : 1.

(xiii) Angle Bisector Theorem–

In the figure, if AD is the angle bisector (interior) of Ð BAC. Then,




1. \(AB/AC = BD/DC.\)
2. \(AB \times AC – BD \times DC = AD^2 \)

(xiv) Midpoint Theorem –

In a triangle, the line joining the mid points of two sides is parallel to the third side and half of it.

(xv) Basic Proportionality Theorem

A line parallel to any one side of a triangle divides the other two sides proportionally. If DE is parallel to BC, then



\(\frac{AD}{BD}=\frac{AE}{EC}, \frac{AB}{AD}=\frac{AC}{AE}, \frac{AD}{DE}=\frac{AB}{BC}\) and so on





Properties of circle

(i) Only one circle can pass through three given points.

(ii) There is one and only one tangent to the circle passing through any point on the circle.

(iii) From any exterior point of the circle, two tangents can be drawn onto the circle.

(iv) The lengths of two tangents segment from the exterior point to the circle, are equal.

(v) The tangent at any point of a circle and the radius through the point are perpendicular to each other.

(vi) When two circles touch each other, their centers & the point of contact are collinear.

(vii) If two circles touch externally, the distance between centers = sum of radii.

(viii) If two circles touch internally, the distance between centers = difference of radii

(ix) Circles with the same center and different radii are concentric circles.

(x) Points lying on the same circle are called concyclic points.

(xi) Measure of an arc means a measure of the central angle.

m(minor arc) + m(major arc) = 360°.

(xii) Angle in a semicircle is a right angle.

(xiii) Only one circle can pass through three given

(xxv) If ON is \(\perp\) from the center O of a circle to a chord AB, then AN = NB



(\(\perp\) from center bisects chord)

(xv) If N is the midpoint of a chord AB of a circle with center O, then ÐONA = 90°.
(Converse, \(\perp\) from center bisects chord)

(xvi) Two congruent figures have equal areas but the converse need not be true.

(xvii) A diagonal of a parallelogram divides it into two triangles of equal area.

(xviii) Parallelograms on the same base and between the same parallels are equal in area.

(xix) Triangles on the same bases and between the same parallels are equal in area.

(xx) If a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is equal to half of the parallelogram.

If PT is tangent to the circle, then \(OP^2= PT^2= OT^2\)



If PT is tangent and PAB is the secant of a circle, then \(PT^2= PA.PB\)




If PB & PD are two secants of a circle, then PA.PB = PC.PD



If two circles touch externally, then the distance between their centers = \((r_1 + r_2)\)



If two circles touch internally, then the distance between their centers \(= r_1– r_2\) where \(r_1> r_2\)