Question

1. Write the coordinates of the alphabet from maple leaf which are present in the word MATHEMATICS.(write only once for repeated alphabet)

2. Find the distance between the points I and M.

3. Find the midpoint of the line segment DP.

4. Find the coordinates of the point which divides the join of C and M in the ratio2:3.​

Answer 1

Question - (1) :-

Write the coordinates of the alphabet from maple leaf which are present in the word MATHEMATICS.(write only once for repeated alphabet).

Answer and solution :-

• To find the coordinates of alphabet from maple leafs at first we have noticed in the given figure. with the help of figure we can easily calculate the coordinates for alphabets from the word (Mathematics).

Alphabet from Mathematics word :-

• As per the given graphic points M is represented in the form x and y axis (x, y) or (x' , y')

Coordinates for M = (5 , 5)

Coordinates for A = (-5 , -4)

Coordinates for T = (2 , -6)

Coordinates for H = (-4 , 8)

Coordinates for E = (-5 , 4)

Coordinates for I = (-2 , 7)

Coordinates for C = (-7 , 2)

Coordinates for S = (1 , -3)

Question - (2) :-

Find the distance between the points I and M.

Answer and solution :-

• To calculate distance between two points l and M at first we have to find coordinates points for l and M from given figure.

Coordinates for l = (-2 , 7)

Coordinates for M = (5 , 5)

• Using distance between two points formula:-

IM = √(x - x')² + (y - y')²

• x = -2. x' = 5. y = 7. y' = 5

IM = √(-2 - 5)² + (7 - 5)²

IM = √(-7)² + (2)² => √49 + 4 = √53 units

Question - (3) :-

Find the midpoint of the line segment DP.

Answer and solution :-

• To calculate the midpoint of linesegment DP at first we have to find out the coordinates points for D and P from the given figure.

Coordinates for D = (-8 , 5)

Coordinates for P = (7 , 2)

• By applying midpoint formula we get :-

Midpoint of linesegment (DP) = (x + x')/2 , (y + y')/2

• x = -8. , x' = 7. y = 5. y' = 2

Midpoints = (-8 + 7)/2 ,. (5 + 2)/2

Midpoints = (-1/2 , 7/2)

Question - (4) :-

Find the coordinates of the point which divides the join of C and M in the ratio2:3.

Answer and solution :-

• To calculate the coordinates points for dividers that join linesegment CM. Let P is the points that divides linesegment in the ratio of 2 : 3. Simply by using section formula we can easily calculate the bisector points of linesegment CM

Coodinates for C = (-7 , 2)

Coordinates for M = (5 , 5)

• By applying section formula we get :-

Coordinates for P(x , y) = (mx' + m'x/m + m') , (my' + m'y/m + m')

• m = 2 , m' = 3 x = -7. , x' = 5 , y = 2 , y' = 5

P(x , y) = (2×5 + 3×(-7)/2 + 3 , (2×5 + 3×2)/2 + 3

P(x, y) = (10 - 21)/5 , (10 + 6)/5

P(x, y) = -11/5 , 16/5

Answer 2

$\quad \qquad { \sf { \huge { \mathfrak { \red { \underline { \underbrace { Question \: 1 } } } } } } }$

We have to Write the coordinates of the alphabet MATHEMATICS from the given figure and only once for each repeated Letter . Which Implies , we have to write the coordinates for each letter of The Word MATHEICS . We first write the abscissa and then ordinates of the corresponding letters in parentheses !

Coordinates of M = ( 5 , 5 )

Coordinates of A = ( - 5 , - 4 )

Coordinates of T = ( 2 , - 6 )

Coordinates of H = ( - 4 , 8 )

Coordinates of E = ( - 5 , 4 )

Coordinates of I = ( - 2 , 7 )

Coordinates of C = ( - 7 , 2 )

Coordinates of S = ( 1 , - 3 )

$\quad \qquad { \sf { \huge { \mathfrak { \red { \underline { \underbrace { Question \: 2 } } } } } } }$

Coordinates of I = ( - 2 , 7 )

Coordinates of M = ( 5 , 5 )

Lets consider two points :-

${ \sf { \bf { ( x_{1} , y_{1} ) \: and \: ( x_{2} , y_{2} ) } } }$

Then the distance between both points is given by :-

${ \sf { \bf { \sqrt{(x_{2} - x_{1} )² + (y_{2} - y_{1})²} } } }$

( You should note that we have to subtract From which point is along to the equal sign Like here is M )So applying same concept we gets ;

${ \sf { IM = { \sqrt{[ 5 - ( - 2 ) ]² + ( 5 - 7 )²} } } }$

$: \longmapsto { \sf { IM = { \sqrt{( 5 + 2 )² + ( - 2 )²} } } }$

$: \longmapsto { \sf { IM = { \sqrt{(7)² + ( - 2 )²} } } }$

$: \longmapsto { \sf { IM = { \sqrt{49 + 4} } } }$

$: \longmapsto { \sf { IM = { \sqrt{53} } } }$

Henceforth , The distance between I and M is √53 units as scale is not given

$\quad \qquad { \sf { \huge { \mathfrak { \red { \underline { \underbrace { Question \: 3 } } } } } } }$

Coordinates of D = ( - 8 , 5 ) Coordinates of P = ( 7 , 2 )

We have to find the Midpoint of a point DP . So we are going to use Midpoint Formula ;

Consider two points ;

${ \sf { \bf { ( x_{1} , y_{1} ) \: and \: ( x_{2} , y_{2} ) } } }$

The Midpoint of the above coordinates is given by :-

${ \sf { \bf { \bigg ( \dfrac{x_{1} + x_{2}}{2} \bigg ) , \bigg ( \dfrac{y_{1} + y_{2}}{2} \bigg ) } } }$

So , Applying this same concept ;

Let , Midpoint of DP = M

${ \sf { Coordinates \: of \: M = \: { \bigg ( \dfrac{- 8 + 7}{2} \bigg ) , \bigg ( \dfrac{5 + 2}{2} \bigg ) } } }$

$: \longmapsto { \sf { Coordinates \: of \: M = \: { \bigg ( \dfrac{- 1}{2} \bigg ) , \bigg ( \dfrac{7}{2} \bigg ) } } }$

Henceforth , The Midpoint of DP is M ( - 1/2 , 7/2 ) .

$\quad \qquad { \sf { \huge { \mathfrak { \red { \underline { \underbrace { Question \: 4 } } } } } } }$

We have to Find the coordinates of The point joining the Points C and M in the ratio 2 : 3 .

We are going to use Section Formula

Consider two points :-

${ \sf { \bf { ( x_{1} , y_{1} ) \: and \: ( x_{2} , y_{2} ) } } }$

The Coordinates of the point which divides the line joining the above points in the ratio m : n is given by:-

${ \sf { \bf { \bigg ( \dfrac{m × x_{2} + n × x_{1}}{m + n}} \bigg ) , { \bigg ( \dfrac{ m × y_{2} + n × y_{1}}{m + n } } \bigg ) } }$

So , applying this same concept ;

Coordinates of C = ( - 7 , 2 )

Coordinates of M = ( 5 , 5 )

Let a , b be the coordinates of the Point Joing C and M dividing it in the ratio 2 : 3 .

By section Formula ;

${ \sf { \bf { ( a , b ) = \bigg ( \dfrac{2 × 5 + 3 × - 7}{2 + 3}} \bigg ) , { \bigg ( \dfrac{ 2 × 5 + 3 × 2}{2 + 3 } } \bigg ) } }$

${ \sf { \bf { ( a , b ) = \bigg ( \dfrac{10 - 21}{5}} \bigg ) , { \bigg ( \dfrac{10+6}{5 } } \bigg ) } }$

${ \sf { \bf { ( a , b ) = \bigg ( \dfrac{-11}{5}} \bigg ) , { \bigg ( \dfrac{16}{5 } } \bigg ) } }$

Henceforth , The Required coordinates are ( -11/5 , 16/5 )