Geometry

Pratyush @ Toughest Questions

Geometry I


Lines


Two intersecting lines q

p r

s

Is q = s?

Yes, Opposite angles

p + q + r + s = ?


Parallel Lines a b

c d

Is a = c?b + c = ?


Triangles


In an isosceles triangle, if two sides are 6 & 12,What is the length of the third side?

12

Sum of 2 sides > third.


In a Triangle the angles are a, b, c (integers) and a > b > cThe maximum value of a?

177 ( b & c = 2 & 1)

The minimum value of a?

61 ( b & c = 60 & 59)


Right Triangles R

S T

(RS)2 + (ST) 2 = (RT)2


Right Isosceles triangle R

S T

Ratio of RS: ST: RT

1 : 1 : 2


30-60-90 Triangle R

S T

R = 60, T = 30.

RS: ST: RT is 1 : 3 : 2


Area = 1/2 x Base x Height

C

B

A

Area of a triangle


Altitude of an equilateral triangle = ?

(3)/2 x side

What is area of this triangle?

(3)/4 * side2


Quadrilaterals


Parallelogram

Area of parallelogram =

Base * Altitude


If the diagonals of a parallelogram are equal, then it is also -A Rhombus

A Rectangle

Answer: Rectangle


Area of a trapezium?

1/2 x (Sum of parallel sides) x height


Polygons


Is this figure a polygon?

No.


Polygons

Sum of interior angles of a Polygon of n sides (and n angles) is

( 2n - 4) x 90


Circles


Circle of radius r

Circumference = ?

Area = ?


What is the ratio of the area of the sector (angle x) to that of the circle?

Ratio of areas = x / 360

R

Ox S


If the radius of the circle is 1, what is the side of the inscribed square?

The diagonal of the square will be 1+1 = 2.

So the side will be 2


If the radius of the circle is 1, what is the side of the circumscribed hexagon?

Draw an equilateral triangle.

Height of triangle = radius = 1

So side of triangle = 2 / 3


What can be said about the triangle inscribed in a semi-circle?

It is always a right triangle


Solids


A rectangular solid has sides 3, 4 and 8. What are

surface areas and volume?

Volume = 3 x 4 x 8 = 96

Area = 136.


A cylindrical solid has base radius r and height h. What is the surface

area and volume?

Volume = r2h

Area = 2(r2) + 2rh


Co-ordinate Geometry


The quadrants

II I

III IV

2

1

3

4

5

-4

-5

-3

-2

-1

y

21 3 4 5-2 -1-3-4-5x


What would be the signs of x & y co-ordinates?

For points in quadrant I(+, +)

For points in quadrant II(-, +)


What would be the signs of x & y co-ordinates?

For points in quadrant III(-, -)

For points in quadrant IV(+, -)


What is distance AB?

A (-2, 4)

B(3,-3)

Horizontal: 5, Vertical: 7

Total Distance = 74

2

1

3

4

5

-4

-5

-3

-2

-1

y

21 3 4 5-2 -1-3-4-5x

III

IIIIV


Are these points on the same line?

A (-2, 2), B(0,1), C(2,0)

If all points satisfy the same eqn, then they are on one line

y = -1/2 x + 1 is the eqn here.


General form of Line equation

y = mx + c

m: the slope, c: the y intercept


Calculation of slope

Slope = (Y2 - Y1) / (X2 - X1)

or, Difference in y coordinates Difference in x coordinates


Given 2 points (x1,y1) & (x2,y2), The equation of

line =

(y - y1)/(x - x1) = (y1 - y2)/(x1 - x2)


For points (-2, 4) & (3,-3), what is the equation?

-7/5 = (y - 4) / (x - (-2))

y = -7/5 x + 6/5


What is the equation of a line..

Parallel to x axis

y = a

parallel to y axis

x = a


To conclude Part I


Lines Opposite angles are equal

Sum of internal angles for parallel lines = 180

Corresponding angles are equal


Polygons

Sum of interior angles for

Polygon of n sides

( 2n - 4) x 90


Quadrilaterals

Area of parallelogram =

Base x Altitude

Area of a trapezium = 1/2xSum of parallel sides x height


Circles

Circumference = 2r

Area = r2

Measure of sector angle = x, Ratio of areas = x / 360


A cylindrical solid has base radius r and height h. What is the surface

area and volume?Volume = r2h

Area = 2(r2) + 2rh


What will be distance between points (x1,y1) &

(x2,y2)?

(x1 - x2)2 + (y1- y2 )2


General form of Line equation

y = mx + c


Calculation of slope

Difference in y coordinates Difference in x coordinates


Given 2 points (x1,y1), (x2,y2), the equation =

(y - y1)/(x - x1) = (y1 - y2)/(x1 - x2)


Geometry II


Find the area of a rectangle, if the ratio of one side to diagonal is 3: 5 and the other side is 8 cm?


This will obviously form a 3:4:5 right triangle, with 5 as the diagonal. If the length of one side is equal to 8, then the diagonal is 10 and the other side is 6. So area = 48 cm2.


The difference between the circumference and the diameter of the circle is 60 cm. The area of the circle isd - d = 60, d = 60 / ( - 1) = 28

Area = d2 / 4 = 616


A room of length 7 m and breadth 5.6 m is to be carpeted leaving 30 cm open space all round. Calculate the carpet area.


length of carpet = 6.4 m. breadth of carpet = 5 m.

Total area of carpet = 32 m2


It costs $ 0.3 / cm2 to carpet a room. The cost of carpeting a room (x/100) meters long and half as broad is ...


x/100 meters long is x cm; and the other dimension will be x/2

cm.

Area of room = x2 / 2 cm2,

cost of carpeting = 0.3x2 / 2

= $ 3x2/20


In a rectangular field of 50mX6m, a tank 4 m long, 3 m broad and 2 m deep is dug. The soil taken out is spread evenly. By how many meters will the level of the field be raised?


Soil taken out from tank = 24m3. Area over which this

earth is spread = area of rectangular field -

area of tank = (50 X 6) - (4 X 3) = 300 - 12 = 288m3. The increase in level of

field = 24/288 = 1/12m


If the volume of a sphere is divided by its surface area, the result is 27 cm. The radius of the sphere is


Volume of sphere Surface area of sphere

= 27 (4/3 r3) / 4 r2 = 27,

r = 81cm.


Two circular cylinders of equal volume have their heights in the ratio 1 : 2. What will be the ratio of their radii?


Volume of cylinder 1 Volume of cylinder 2

= 1 Therefore r1

2h1 / r22h2 = 1 i.e.

r12 / r2

2 = 2 , r1/r2 = ( 2:1)


One side of a rectangle is x inches. If the perimeter is p inches, what is the length ( in inches) of the other side?


Perimeter = sum of the sides.

So, p = 2x + 2 (other side) Other side = (p -2x) / 2


Calculate the area of triangle ABC where AB is the diameter of a circle of radius 5 and C is on that circle at a distance of 6 units from B.


AB is the diameter and C lies on the circle. Hence, angle ACB = 900. (Any triangle

inscribed in a semi-circle is always a right triangle.) By Pythagorus theorem, AC = 8. Area = 1/2 AC. BC = 1/2 x 8 x 6 = 24.


Volume of a cylindrical room surmounted by a hemispherical dome is 28.6m3. Radius of the room = Height of the cylindrical part. Find the height of the room.


volume = (of cylinder ) + (of hemisphere)

= 28.6 i.e r2 *r + 2/3 r3 = 28.6 i.e. 5/3 * 22/7 * r3 = 28.6

i.e. r = 1.75. Height of room = 3.5m


A spherical metal ball 3cm in diameter is melted into three spherical balls. The radii of two balls are 0.75 cm and 1.25 cm. What is the radius of the 3rd ball?


Volume of 1st ball = sum of the volumes of each of 3 spherical

balls4/3 (1.5) 3 = 4/3 [(0.75) 3 +

(1.25) 3 + x3]x3 = (1.5) 3 - (0.75) 3 - (1.25) 3

= 1


3 vertices of an equilateral triangle with side 21 cm are centers of 3 circles touching each other. What is the area of the triangle that is not covered by the three circles?


The area of the required part =( 3/4 a2) - 3( r 2/ 6)= ( 3/4 a2)- ( r 2/ 2)

= a2( 2 3 - ) / 8 = 17.8


Three cubes of metal with edges in the ratio 3 : 4 : 5 are melted into one cube, with diagonal 48 3 m. Calculate the edges of the three cubes.


Total volume of the three cubes = 27x3 + 64x3 + 125x3 = 216x3. The length of side of the new cube = diagonal / 3 = 48m. Volume of cube = 48 * 48 * 48 i.e. 216x3 = (48) 3 gives = (6x)3 = (48)3, 6x = 48, x = 8. The length of edges of the cube are 24m, 32m and 40m

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