**SSC CHSL Topic Wise Study Material – Quantitative Aptitude – Geometry**

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Geometry deals with the study of shape and size of 2-dimensional and 3-dimensional figures. But here, we will study about the plane geometry i.e., 2-dimensional figures only.

**Basic Definitions**

**Point** It is defined by its position having no length, width or thickness.

**Line** A geometrical line is a set of points that extends endlessly in both the directions.

**Parallel** **Lines** Two lines in the same plane are called parallel, if they never meet, although, they are extended in either direction. They remain at same distance for the whole length. Parallel lines are denoted by the symbol ‘||’

A line which cuts a pair of parallel lines is called a transversal. In the figure, AB and CD are parallel lines and EF is the transversal.

**Intersecting Lines**

If two lines have a point in common, they are said to be intersecting lines. Two lines can intersect at the most at one point.

**Collinear Points**

Three or more than three points are said to be collinear if there is a line which contains them all.

**Concurrent Lines**

Three or more than three points are said to be concurrent if there is a point which lies on all of them.

**Angles** When two line segments intersect, then the point of intersection form an angle.

Here, AOB is an angle

**Types of Angles**

**(i) Acute Angle** The angle whose value lies between 0° and 90° is called an acute angle, (where, 0° < a < 90°)

**(ii) Right Angle** The angle whose value is 90° is called right angle, (where, c = 90°)

**(iii) Obtuse Angle** The angle whose value lies between 90° and 180° is called an obtuse angle, (where, 90° < b < 180°)

**(iv) Straight Angle** The angle whose value is 180°. Its called a straight angle, (where, d = 180°)

**(v) Reflex Angle** The angle whose value lies between 180° and 360° is called reflex angle, (where, 180°<e<360°)

**(vi) Complete Angle** The angle whose value is 360° is called a complete angle.

**(vii) Complementary Angle** If the sum of two angles is 90°,then they are called complementary angles to each Other.

**(viii) Supplementary Angle** If the sum of two angles is 180° then, they are called supplementary angles.

Types of angle formed a parallel line due to a transversal

• Corresponding angles are equal

∠1=∠5, ∠2 =∠6

∠4=∠8,∠3=∠7

• Alternate interior angles are equal

∠3 = ∠5

∠4 = ∠6

• Vertically opposite angles are equal

∠2 =∠4, ∠1 = ∠3, ∠5 = ∠7,∠6 = ∠8

**Angle Bisectors** A line which cuts an angle into two equal angles is called an angle bisector.

**Types of Angle bisector**

**(i) Internal Angle Bisector** Here, two angles are formed ∠AOB and ∠BOC. Both angles are equal (θ) because OB is the internal bisector.

**(ii) External Angle Bisector** Here, ∠A’OB and ∠BOC are equal and external bisector is OB.

**Polygons**

A polygon is a closed shape bounded by a number of line segments.

The line segments are called its sides and the points of intersection of consecutive sides are called its vertices. An angle formed by two consecutive sides of a polygon is called its interior angle or simply an angle of the polygon.

A polygon is named according to the number of sides, it has

**Convex Polygon** A polygon in which none of Its interior angles is more than 180° is called convex polygon.

**Concave Polygon** A polygon in which atleast one angle is more than 180° is called concave polygon.

**Regular Polygon** A regular polygon has all its sides and angles equal.

**(i)** Each exterior angle of a regular polygon = 360°/Number of sides

**(ii)** Each interior angle = 180° – exterior angle.

In a convex polygon of n sides, we have

(a) Sum of all interior angles = (2n – 4) x 90°

(b) Sum of all exterior angles = 360°

(c) Number of diagonals of polygon on n sides =n(n-3/2)

**Triangle**

A plane figure bounded by three lines in a planes is called a triangle.

A triangle has three vertices A,B and C, three angles ∠A, ∠B and ∠C, three sides AB, BC and AC.

**Types of Triangles**

**(i) Equilateral Triangle** A triangle having all sides equal is called an equilateral triangle.

**(ii) Scalene Triangle** A triangle having all sides of different length us called a scalene triangle.

**(iii) Isosceles Triangle** A triangle having two sides equal is called an isosceles triangle.

**(iv) Right angled Triangle** A triangle one of whose angles measures 90° is called a right angled triangle.

**(v) Acute angled Triangle** A triangle whose each angle is acute, is called and acute angled triangle.

**(vi) Obtuse angled Triangle** A triangle with one angle is an obtuse angle, is called an obtuse angled triangle.

**Some Important Terms Related to a Triangle**

**Median** The median of a triangle corresponding to any side is the line segment joining the mid point of that side with the opposite vertex

In the figure given below, AD, BE and CF are the medians.

The medians of a triangle are concurrent ie., they intersect each other at the same point.

**Centroid** The point of intersection of all the three medians of a triangle is called its centroid.

In the above figure G is the centroid of ΔABC.

Note The centroid divides a median in the ratio 2:1.

**Altitudes** The altitude of a triangle corresponding to any side is the length of perpendicular drawn from the opposite vertex to that side.

In the figure given above, AL, BM and CN are the altitudes.

**Note** The altitudes of a triangle are concurrent.

**Orthocentre** The point of intersection of all the three altitudes of a triangle is called its orthocentre

In the figure given above H is the orthocentre of ΔABC.

Note The orthocentre of a right-angled lies at the vertex containing the right angle.

**Incentre of a Triangle** The point of intersection ,of the internal bisectors of the angles of a triangle is called its incentre. In the figure given below, the internal bisectors of the angles of ΔABC intersect at I.

I is the incentre of ΔABC Let ID⊥BC

Then, a circle With centre I and radius ID is called the in circle of ΔABC.

**Note** The incentre of a triangle is equidistant from is sides.

**Circumcentre of a Triangle** The point of intersection of the perpendicular bisectors of the sides of a triangle is called its circumcentre.

In the figure given below, the right bisectors of the sides of ΔABC intersect at 0.

O is the circumcentre of ΔABC with 0 as centre and radius equal to OA = OB = OC. We draw a circle passing through the vertices of the given Δ.

This circle is called the circumcircle of ΔABC.

Note The circumcentre of a triangle is equidistant from its vertices.

**Example An equilateral Δ TQR is drawn inside a square PQRS. The value of the ∠PTS in degrees, is SSC (10 + 2) 2012**

(a) 75°

(b) 90°

(c)120°

(d)150°

**Congruency of Triangle**

Two triangles are congruent, if

(i) Three sides of one triangle are respectively equal to three sides of the other (SSS).

(ii) Two sides and the angle made by them, of the one triangle are equal to the corresponding two sides and the angle made by them of the other (SAS).

(iii) Two angle and a side of one triangle are equal to the corresponding two angles and a side of the other (ASA).

**Similarity of Triangle**

Two triangles are said to be similar, if their corresponding angles are equal and their corresponding sides are proportional i.e., they have the same shape but may have different sizes.

(i) Ratio of the area of two similar triangles is equal to the ratio of the squares of any two corresponding sides.

(ii) Ratio of the areas of two similar triangles is equal to the ratio of the squares of corresponding altitudes and medians.

(iii) The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of sides containing the angle.

(iv) The lines joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.

**Example The side AB of a parallelogram ABCD is produced to E in such way that BE = AB. DE intersects BC at Q. The point Q divides BC in the ratio SSC (10+2) 2012**

(a) 1 : 2

(b) 1 : 1

(c) 2 : 3

(d) 2 : 1

**Circle**

A circle is a set of points which are equidistant from a given fixed point. The fixed point is known as the centre of that circle and the given distance is called the radius of the circle.

Here, ‘O’ is the centre of the circle and V is the radius.

**Chords in a Circle**

(i) A straight line drown from the centre of a circle to bisect a chord which is not a diameter is at right angle to the chord.

(ii) Equal chords of a circle are equidistant from the centre. Conversely, the chords that are equidistant from the centre are equal.

(iii) If two chords say AB and CD of a circle intersect each other internally or externally at point E, then.

AE x EB = DE x EC

**Example If the chord of a circle of radius 5 cm is a tangent to a circle of radius 3 cm, both the circles being concentric, then the length of the chord is SSC (10 + 2)2011**

(a) 10 cm

(b) 12.5 cm

(c) 8 cm

(d) 7 cm

**Example The length of two chords AB and AC of a circle are 8 cm and 6 cm and ∠BAC = 90°, then the radius of circle is ssc (10+2) 2011**

(a) 25 cm

(b) 20 cm

(c)4cm

(d)5 cm

**Tangents to a Circle**

(i) The tangent at any point of a circle is perpendicular to the radius through the point of contact i.e., OT⊥PT.

(ii) If two tangents are drawn to a circle from an outside point, the length of the tangent from the external point to their respective points of contact are equal i.e.,PA=PB

(iii) If PT is tangent (with P being an external point and T being the point of contact) and PAP is a secant to circle (with A and B as the points where the recant cuts the circle), then PT² = PA x PB

**Pairs of Circles**

(i) In a given pair of circles, there are two types of tangents. The direct tangents and the cross (or transverse) tangents. In the figure, AB and CD are the direct tangents and EH and GF are the transverse tangent.

(ii) When two circles of radii r1 and r2 have their centres at a distance d, the length of the direct common tangent = √d²-(r1 – r2)² and the length of transverse tangent = √d²-(r1 + r2)² . If the two circles touch, then d = r1+r2

**Example The distance between the centres of two equal circles, each of radius 3 cm, is 10 cm. The length of a transverse common tangent is SSC (10+2) 2012**

(a) 4 cm

(b) 6 cm

(c) 8 cm

(d) 10 cm

**Reference Corner**

**1. The point P(a, b) is first reflected in origin to and PI is reflected in y-axis to (5, -4). The coordinates of point P are SSC (10 + 2) 2017**

(a)(-5,-4)

(b)( -5,4)

(c)(5, 4)

(d)(5, -4)

**Answer:**

(b)

**2. D and E are mid-points of sides AB and AC respectively of ΔABC. A line drawn from A meets BC at H and DE at K,AK: KH =?:?**

(a) 1:2

(b) 2:1

(c) 1:1

(d) 1:3

**Answer:**

**3. In ΔABC, ∠A = ∠B = 60°, AC = √13 cm. The lines AD and BD intersect at D with ∠D = 90°. If DB = 2 cm, then the length of AD is SSC (10 + 2) 2014**

(a) 4 cm

(b)4.7cm

(c) 3 cm

(d)3.5cm

**Answer:**

**4. In ΔABC, the medians AD, BE and CF intersect each other at the point G. If the area of ΔABC is 36 sq cm, then the area (in sq cm) of the quadrilateral BDGF is equal to SSC (10 + 2) 2014**

(a) 18

(c) 6

(b) 24

(d) 12

**Answer:**

**5. In ΔABC, D is the mid-point of BC. Length of AD is 27 cm. N is a point on AD such that the length of DN is 12 cm. The distance of N from the centroid of ΔABC is equal SSC (10+2) 2014**

(a) 9 cm

(b) 15 cm

(c) 3 cm

(d) 6 cm

**Answer:**

**6. The side BC of a ΔABC is extended to D. If ∠ACD = 120° and∠ABC = 1/2∠CAB, then the value of ∠ABC is SSC (10 + 2) 2014**

(a) 60°

(b) 20°

(c) 80°

(d) 40°

**Answer:**

**7.The side QR of an equilateral triangle PQR is produced to the point S in such a way that QR = RS and P is joined to S. Then, the measure of ∠PSR is SSC (10 + 2) 2013**

(a) 60°

(b) 45°

(c) 30°

(d) 15°

**Answer:**

**8.PQ is a chord of length 8 cm, of a circle with centre O and of radius 5 cm. The tangents at P and Q intersect at a point T. The length of TP is SSC (10+2) 2013**

(a)10/3 cm

(b)15/4 cm

(c)20/3 cm

(d)21/4 cm

**Answer:**

**9.If a chord of a circle is equal to the radius of the circle, then the angle subtended by the chord at a point on the minor arc is SSC (10 + 2) 2013**

(a) 120°

(b) 30°

(c) 150°

(d) 60°

**Answer:**

**10.The sum of interior angles of a regular polygon is 1440°. The number of sides of the polygon is SSC (10 + 2) 2013**

(a) 6

(b) 8

(c) 10

(d) 12

**Answer:**

**Practice Exercise**

**1.In the adjoining figure AB||CD and PQ, QR intersect AB and CD both at E, F, G and H respectively. Given that m ∠PEB = 80°, m ∠QHD = 120° and m ∠PQR = x°, find the value of x.**

(a) 40°

(b) 20°

(c) 100°

(d) 30°

**2.Two parallel lines AB and CD are intersected by a transversal EF at M and N respectively. The lines MP and NP are the bisectors of interior angle ∠BMN and ∠DNM on the same side of the transversal. Then, ∠MPN is equal to**

(a) 60°

(b) 90°

(c) 45°

(d) 120°

**3.In the given figure, ΔABC is an isosceles triangle in which AB = AC and ∠ABC = 50°, then ∠BDC is equal to**

(a) 110°

(b) 100°

(c) 90%

(d) 80°

**4.In the given figure, PT is the tangent of a circle with centre O at point R. If diameter SQ is increased, it meets with PT at point P. If ∠SPR=x° and ∠QSR= y°, what is the value of x° + 2y° ?**

(a) 90°

(b) 105°

(c) 135°

(d) 180°

**5.If (5y + 62°), (22° +y) are supplementary, find Y**

(a) 16°

(b) 32°

(c) 8°

(d) 1°

**6.In the liven diagram AB|| GH||DEand GF||BD||HI, ∠FGC = 80°, find the value of ∠CHI.**

(a) 80°

(b)120°

(c)100°

(d)160

**7. In the given figure, ∠PQR = 90°. O is the centroid of the ΔPQR, where PQ = 5 cm and QR = 12 cm, then OQ is equal to**

(a)3 1/2cm

(b)4 1/3cm

(c)4 1/2cm

(d)5 1/2cm

**8. ΔABC is such that AB = 9 cm, BC = 6 cm, AC = 7.5 cm. ΔDEF is similar to ΔABC. If EF = 12 cm, then DE is**

(a) 6 cm

(b) 16 cm

(c) 18 cm

(d) 15 cm

**9.In the adjoining figure of ΔABC, ∠BCA = 120° and AB = c, BC = a, AC = b**

(a) c² = a² + b² + ab

(b) c² = a² + b² -ba

(c) c² = a² + b² – 2abc

(d)c² =a² + b² + 2ab

**10.If AB || CD and the line EF cuts these lines at the points M and N, respectively. Bisectors of angles ∠BMN and ∠MND meet at the point Q. Find the value of ∠MQN.**

(a)90°

(b)80°

(c)70°

(d)60°

**11. A ΔPQR is formed by joining the mid-points of the sides of at ΔABC. ‘O’ is the circumcentre of ΔABC, then for ΔPQR, the point ‘O’ is**

(a) Incentre

(b) Circumcentre

(c) Ortho centre

(d) Centroid

**12. The triangle is formed by joining the mid-points of the sides AB,BC and CA of ΔABC on the area of Δ PQR is 6 cm², then the area of ΔABC is**

(a) 36 cm²

(b) 12 cm²

(c) 18 cm²

(d) 21 cm²

**13. In the given figure, ABCD is a parallelogram. E and F are the centroids of ΔABD and ΔBCD, respectively, EF is equal to**

(a)BE

(b)AE

(c)CE

(d)DE

**14. If O is the center of the circle, find the value of x in the given figure.**

(a) 50°

(b) 60°

(c) 70°

(d) 80°

**15.In the figure BC||AD. find the value of x.**

(a) 9,10

(b) 7,8

(c) 10,12

(d) 8,9

**16.In the given figure, ∠BAC and ∠BDC are angles of some segments. ∠DBC = 30° and ∠BCD = 110°. Find m ∠BAC is**

(a) 35°

(b) 40°

(c) 55°

(d) 60°

**17. In the given figure, ∠COE = 90°. Find the value of x.**

(a) 120°

(b) 60°

(c) 45°

(d) 30°

**18.In Δ ABC, DE||BC. AD = 2.5 cm, DB = 5 cm, AE = 2 cm and BC = 9 cm. Find EC and DE, respectively.**

(a) 4 cm and 3 cm

(c) 2 cm and 4 cm

(b) 5 cm and 3 cm

(d) 4 cm and 5 cm

**19.O is the incenter of ΔABC and ∠A = 30°, then ∠BOC is SSC (10 + 2) 2011**

(a) 100°

(b) 105°

(c) 110°

(d) 90

**Answers**

**Hints & Solutions**

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