Tips on Solving Questions Based on Geometry

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Cracking Aptitude Questions on Geometry
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Part I Triangles Angles and Sides

• The sum of any two sides of a triangle is
  greater than the third side.
• The line joining the mid-point of a side of a
  triangle to the opposite vertex is called the
  median. The median of a triangle divides it into
  two triangles of the same area.
• The point of intersection of the 3 medians of a
  triangle is called its centroid. The centroid
  divides each of the medians in the ratio 2 1.
• The perimeter of a triangle of sides of length a,
  b and c is (a b c).
• Question In triangle PQR length of the side QR
  is less than twice the length of the side PQ by 2
  cm. Length of the side PR exceeds the length of
  the side PQ by 10 cm. The perimeter is 40 cm. The
  length of the smallest side of the triangle PQR
  is
• Solution
• QR 2PQ 2 and PR PQ 10PQ QR PR 40
• PQ 2PQ - 2 PQ 10 40
• 4PQ 32 PQ 8 cm which is the smallest side
  of the triangle.
• Question In an isosceles triangle ABC,? A 900,
  AL is drawn perpendicular to BC. Find ? BAL.
• Solution
• ?A ?B ?C 1800
• ?B ?C (1800 900) / 2 450
• ?BAL ?BLA ?B 1800
• ?BAL 1800 900 450 450

A
B
C
L
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Part I Triangles Area

• Area of a triangle Base x Height / 2.
• Area of a triangle is vs(s a)(s b)(s c)
  where s semi perimeter (a b c) / 2.
• The area of the triangle formed by joining the
  mid-points of the sides of a given triangle is
  one-fourth of the area of the given triangle.
• Question Triangle ABC is inscribed in a square
  of side 20 cm. Find the area of the triangle.
• Solution
• From the figure, we can see that
• Base 20 cm, Height 20 cm.
• Thus, area of the triangle 20 x 20 / 2 200
  cm2
• Question What is the height of the triangle?
• I. The area of the triangle is 20 times its base.
  
• II. Perimeter of the triangle Perimeter of a
  square of side 10 cm.
• I alone sufficient while II alone not sufficient
  to answer
• II alone sufficient while I alone not sufficient
  to answer
• Either I or II alone sufficient to answer
• Both I and II are not sufficient to answer
• Both I and II are necessary to answer

A
20 cm
B
C
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Part I Triangles Types and Properties

• Equilateral Triangle
• The angles of the triangle are equal to 60 each.
  
• All the sides of the triangle are equal.
  
  
• Altitude of an equilateral triangle v3a / 2
• Area of an equilateral triangle v3a2/4
• Isosceles Triangle
• The two base angles of the triangle are equal.
  
• Two sides of the triangle are equal.
  
• Scalene Triangle
• No two angles have the same value.
  
• No two sides have the same length.
  
• Acute-angled Triangle
• All angles of the triangle are lesser than 90.

a
h
Equilateral Triangle
a
h
b
Isosceles Triangle
b
a
c
Scalene Triangle/ Acute Angled Triangle
a
b
c
Right-Angled Triangle
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Part II Quadrilaterals Types
Trapezium A quadrilateral having exactly one
pair of parallel sides is called a trapezium.
Isosceles Trapezium A trapezium whose
non-parallel sides are equal in length is called
an isosceles trapezium. Parallelogram A
quadrilateral is said to be a parallelogram if
both pairs of its opposite sides are parallel.
Rhombus A parallelogram having all sides
equal is called a rhombus. Rectangle A
quadrilateral in which each angle is a right
angle is called a rectangle. Square A square
is a quadrilateral in which all sides are equal
and each angle measures 90.
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Part II Quadrilaterals Angles and Sides

• The diagonals of a parallelogram bisect each
  other.
• Perimeter of a rectangle 2(Length Breadth)
• Perimeter of a square 4 x length of a side
• Each diagonal of a parallelogram divides it into
  triangles of the same area.
• The diagonals of a rectangle are equal and
  bisect each other.
• The diagonals of a square are equal and bisect
  each other at right angles.
• The diagonal of a square of side a av2.
• The diagonals of a rhombus are unequal and
  bisect each other at right angles.
• The sum of angles of a quadrilateral is 360
• Question What is the smaller angle of a
  parallelogram?
• I. Ratio of the angles of a triangle is 345 and
  the larger angle of the parallelogram is 340
  greater than the largest angle of the triangle.
• II. Larger angle of the parallelogram is 38 more
  than its smaller angle.
• I alone sufficient while II alone not sufficient
  to answer
• II alone sufficient while I alone not sufficient
  to answer
• Either I or II alone sufficient to answer
• Both I and II are not sufficient to answer
• Both I and II are necessary to answer

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Part II Quadrilaterals Area

• Area of a rectangle (Length x Breadth).
• Area of a square (side)2 ½ x (diagonal)2.
• Area of parallelogram (Base x Height).
• Area of a rhombus ½ x (Product of diagonals).
• Area of a trapezium ½ x (sum of parallel
  sides) x distance between them.
• Question An error of 2 in excess is made while
  measuring the side of a square. Calculate the
  percentage of error in the calculated area of the
  square.
• Solution
• Let the actual length of side be x. Then measured
  length 1.02x.
• Actual area x2. Measured area (1.02x)2.
• Error in area calculation (1.022 12)x2 / x2
  (1.02 1)(1.02 1) 2.02 x 0.02 0.0404
• Percentage error 0.0404 x 100 4.04
• Question A rectangular park 60 m long and 40 m
  wide has two concrete crossroads running in the
  middle of the park and rest of the park has been
  used as a lawn. If the area of the lawn is 2109
  sq. m, then what is the width of the road?
• Solution
• Let the width of the road be x meters.
• Then, (60 2x)(40 2x) 2109

60 m
60 m
2109 sq. m
40 m
40 m

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Part III Circles Angles, Radii and Area

• Area of a circle pR2, where R is the radius
• Circumference of a circle 2pR
• Length of an arc 2pR? / 360 , where ? is the
  central angle.
• Area of a sector ½ x (arc length x R) pR2? /
  360
• Area of semi-circle pR2 / 2
• Question A lawn is in the shape as shown.
• Find the area of the lawn.
• Solution
• The lawn is in the form of a rectangle with two
  semicircles on opposite ends of the width.
• Area of the rectangle 200m x 14m 2800 sq. m
• Radius of the semicircles 14/2 m 7 m
• Area of the two semicircles 2 x (pR2 / 2) 22
  / 7 x 7 x 7 154 sq. m.
• Total Area of the lawn 2954 sq. m.
• Question ABCD is a square with one vertex at the
  center of the circle and two vertices on the
  circle. What is the length of the diagonal of the
  square if the area of the circle is 100 square
  cm?

200m
14m
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