Geometry Quick Refresh, Exercises of Mathematics

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Vol. 2: Geometry

present

Lines and Angles

Sum of the angles in a straight line is 180° Vertically opposite angles are congruent (equal). If any point is equidistant from the endpoints of a segment, then it must lie on the perpendicular bisector When two parallel lines are intersected by a transversal , corresponding angles are equal, alternate angles are equal and co-interior angles are supplementary. ( All acute angles formed are equal to each other and all obtuse angles are equal to each other ) Tip : The ratio of intercepts formed by a transversal intersecting three parallel lines is equal to the ratio of corresponding intercepts formed by any other transversal.

𝑎𝑎 𝑏𝑏

𝑐𝑐 𝑑𝑑

𝑒𝑒 𝑓𝑓

A Median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. The three medians intersect in a single point, called the Centroid of the triangle. Centroid divides the median in the ratio of 2: An Altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side or an extension of the opposite side. The three altitudes intersect in a single point, called the Orthocenter of the triangle. A Perpendicular Bisector is a line that forms a right angle with one of the triangle's sides and intersects that side at its midpoint. The three perpendicular bisectors intersect in a single point, called the Circumcenter of the triangle. It is the center of the circumcircle which passes through all the vertices of the triangle. An Angle Bisector is a line that divides the angle at one of the vertices in two equal parts. The three angle bisectors intersect in a single point, called the Incenter of the triangle. It is the center of the incircle which touches all sides of a triangle.

Tip : Centroid and Incenter will always lie inside the triangle.

• For an acute angled triangle , the Circumcenter and the Orthocenter will lie inside the triangle.
• For an obtuse angled triangle , the Circumcenter and the Orthocenter will lie outside the triangle.
• For a right angled triangle the Circumcenter will lie at the midpoint of the hypotenuse and the Orthocenter will lie at the vertex at which the angle is 90°.

Tip : The orthocenter , centroid , and circumcenter always lie on the same line known as Euler Line.

• The orthocenter is twice as far from the centroid as the circumcenter is.
• If the triangle is Isosceles then the incenter lies on the same line.
• If the triangle is equilateral , all four are the same point.

Theorems

Mid Point Theorem : The line joining the midpoint of any two sides is parallel to the third side and is half the length of the third side.

Basic Proportionality Theorem : If DE || BC, then AD/DB = AE/EC

Apollonius’ Theorem : AB 2 + AC^2 = 2 (AD 2 + BD 2 )

Interior Angle Bisector Theorem : AE/ED = BA/BD

Continued >> Continued >>

30°-60°-90° Triangle

Area =

√ 2

• x

2

45°-45°-90° Triangle

Area = x

2

30°-30°-120° Triangle

Area =

√ 4

* x

2

Continued >>

Similarity of Triangles

Two triangles are similar if their corresponding angles are congruent and corresponding sides are in proportion.

Tests of similarity : (AA / SSS / SAS)

For similar triangles, if the sides are in the ratio of a:b  Corresponding heights are in the ratio of a:b  Corresponding medians are in the ratio of a:b  Circumradii are in the ratio of a:b  Inradii are in the ratio of a:b  Perimeters are in the ratio of a:b  Areas are in the ratio a 2 : b^2

Congruency of Triangles

Two triangles are congruent if their corresponding sides and angles are congruent.

Tests of congruence : (SSS / SAS / AAS / ASA)

All ratios mentioned in similar triangle are now 1:

If all vertices of a quadrilateral lie on the circumference of a circle, it is known as a cyclic quadrilateral. Opposite angles are supplementary

Area = (^) �(𝑠 − 𝑎𝑎)(𝑠 − 𝑎𝑎)(𝑠 − 𝑎𝑎)(𝑠 − 𝑎𝑎) where s is the

semi perimeter 𝑠 = 𝑎𝑎+𝑏𝑏+𝑐𝑐+𝑑𝑑 2

Parallelogram

Opposite sides are parallel and congruent.

Opposite angles are congruent and consecutive angles are supplementary.

Diagonals of a parallelogram bisect each other.

Perimeter = 2(Sum of adjacent sides); Area = Base x Height = AD x BE

Tip : Sum of product of opposite sides = Product of diagonals

Tip: If a circle can be inscribed in a quadrilateral, its area is given by = √𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

Tip : A parallelogram inscribed in a circle is always a Rectangle. A parallelogram circumscribed about a circle is always a Rhombus.

Continued >>

Rhombus

A parallelogram with all sides equal is a Rhombus. Its diagonals bisect at 90°. Perimeter = 4a; Area = ½ d 1 d 2 ; Area = d x

�𝑎𝑎 2 − (𝑑𝑑 2 )^2

Rectangle

A parallelogram with all angles equal (90°) is a Rectangle. Its diagonals are congruent. Perimeter = 2(l+b); Area = lb

Square

A parallelogram with sides equal and all angles equal is a square. Its diagonals are congruent and bisect at 90°. Perimeter = 4a; Area = a^2 ; Diagonals = a√ 2

Tip : Each diagonal divides a parallelogram in two triangles of equal area.

Tip : Sum of squares of diagonals = Sum of squares of four sides

 AC^2 + BD 2 = AB 2 + BC^2 + CD 2 + DA 2

Tip : A Rectangle is formed by intersection of the four angle bisectors of a parallelogram.

Tip : From all quadrilaterals with a given area, the square has the least perimeter. For all quadrilaterals with a given perimeter, the square has the greatest area. Continued >>

Isosceles Trapezium

The non-parallel sides (lateral sides) are equal in length. Angles made by each parallel side with the lateral sides are equal.

Tip : If a trapezium is inscribed in a circle, it has to be an isosceles trapezium. If a circle can be inscribed in a trapezium, Sum of parallel sides = Sum of lateral sides.

Continued >>

Hexagon (Regular)

Perimeter = 6a; Area =

3√ 2

x a

2

Sum of Interior angles = 720°.

Each Interior Angle = 120°. Exterior = 60°

Number of diagonals = 9 {3 big and 6 small}

Length of big diagonals (3) = 2a

Length of small diagonals (6) = (^) √ 3 a

Area of a Pentagon = 1.72 a^2

Area of an Octagon = 2(√ 2 + 1) a 2

Tip : A regular hexagon can be considered as a combination of six equilateral triangles. All regular polygons can be considered as a combination of ‘n’ isosceles triangles.

Properties of Tangents, Secants and Chords

The radius and tangent are perpendicular to each other. There can only be two tangents from an external point, which are equal in length PA = PB

PA x PB = PC x PD θ = ½ [ m(Arc AC) – m(Arc BD) ]

PA x PB = PC x PD θ = ½ [ m(Arc AC) + m(Arc BD) ]

Continued >>

Properties (contd.)

PA x PB = PC^2 θ = ½ [ m(Arc AC) - m(Arc BC) ]

Alternate Segment Theorem

The angle made by the chord AB with the tangent at A (PQ) is equal to the angle that it subtends on the opposite side of the circumference.  ∠ BAQ = ∠ ACB

Continued >>

Solid Figures

Volume Total Surface

Area

Lateral / Curved

Surface Area

Cube Side

3

6 x Side

2

4 x Side

2

Cuboid L x B x H^ 2(LB + LH +

BH)

2 (LH + BH)

Cylinder 𝜋r

2

h 2 𝜋r (r + h) 2 𝜋rh

Cone (1/3)^ 𝜋r

2

h 𝜋r (r +L) 𝜋rl {where L =

√𝑟^2 + ℎ^2 }

Sphere (4/3)^ 𝜋r

3

4 𝜋r

2

4 𝜋r

2

Hemisphere (2/3)^ 𝜋r

3

3 𝜋r

2

2 𝜋r

2

Tip : There are 4 body diagonals in a cube / cuboid of length (√ 3 x side) and √𝑙𝑙^2 + 𝑎𝑎^2 + ℎ^2 respectively.

Continued >>

Frustum / Truncated Cone

It can be obtained by cutting a cone with a plane parallel to the circular base.

Volume = 1/3 𝜋h (R 2 + r 2 + Rr)

Lateral Surface Area = 𝜋 (R+r) L

Total Surface Area = 𝜋 (R+r) L + 𝜋 (R 2 +r 2 )

Prism

It is a solid with rectangular vertical faces and bases as congruent polygons (of n sides). It will have ‘2n’ Vertices; ‘n+2’ Faces and ‘3n’ Sides / Edges.

Lateral Surface Area = Perimeter x Height

Total Surface Area = Perimeter x Height + 2 Area (^) Base

Volume = AreaBase x Height

Continued >>