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Eduardwww [97]
2 years ago
15

Please help! find the distance of AG if A (4,4) and G (-1,-1)

Mathematics
2 answers:
Naya [18.7K]2 years ago
8 0

  • The distance between two points \rm{A(x_1,y_1)} and \rm{B(x_2,y_2)} is given by the formula,

\rm{AB= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }

Proof:

\rm{Let  \: X'OX \:  and  \: YOY' \:  be \:  the \:  z-axis  \: and \:  y-axis  \: respectively. Then, O \:  is \:  the \:  origin.}

\rm{Let  \: A(x_1,y_1)  \: and  \: B(x_2,y_2)  \: be \:  the \:  given \:  points.}

\rm{Draw \:  AL \perp \: OX, BM \perp \: OX  \: and  \: AN \perp BM }

Now,

\rm{OL=x_1,OM=x_2,AL=y_1  \: and \:  BM=y_2}

\rm\therefore{AN=LM=(OM-OL)=(x_2-x_1)}

\:  \:  \:  \:  \rm{BN=(BM-NM)=(BM-AL)=(y_2-y_1)}

\rm{In  \: right  \: angled \:  \triangle ANB, by  \: Pythagorean  \: theorem,}

We have,

\:  \:  \:  \:  \rm{AB^2=AN^2+BN^2}

\rm{or,AB^2=(x_2-x_1)^2+(y_2-y_1)^2}

\rm\therefore AB= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

The Given question,

Find the distance of AG if A (4,4) and G (-1,-1).

Solution,

The given points are A(4,4) and G(-1,-1).

Then,

\rm{(x_1=4,y_1=4) and (x_2=-1,y_2=-1)}

We know that,

The distance formula,

\rm{AG=  \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}  }

\:  \:  \:  \:  \:  \:  \:  \:  =  \sqrt{ {( - 1 - 4)}^{2}  +  {( - 1  - 4)}^{2} }

\:  \:  \:  \:  \:  \:  \:  \:   =  \sqrt{ {( - 5)}^{2}  +  {( - 5)}^{2} }

\:  \:  \:  \:  \:  \:  \:  \:  =  \sqrt{25 + 25}

\:  \:  \:  \:  \:  \:  \:  \:   =  \sqrt{50}

\:  \:  \:  \:  \:  \:  \:  \:   =  \sqrt{(2)(25)}

\:  \:  \:  \:  \:  \:  \:  \:    =  \sqrt{(2)(5)(5)}

\:  \:  \:  \:  \:  \:  \:  \:    =  \sqrt{(2)( {5}^{2}) }

\rm  \:  \:  \:  \:  \:  \:  \:  \:  = 5  \sqrt{2}  \: units

Answered by:

\frak{\red{moonlight123429}}

DochEvi [55]2 years ago
3 0

{\qquad\qquad\huge\underline{{\sf Answer}}}

Let's solve ~

By using distance formula :

\qquad \sf  \dashrightarrow \:  \sqrt{(x_2 - x_1) {}^{2} + (y_2 - y_1) {}^{2}  }

\qquad \sf  \dashrightarrow \:  \sqrt{(4 - ( - 1)) {}^{2} + (4 - ( - 1)) {}^{2}  }

\qquad \sf  \dashrightarrow \:  \sqrt{(4  + 1) {}^{2} + (4  + 1) {}^{2}  }

\qquad \sf  \dashrightarrow \:  \sqrt{2(5) {}^{2} }

\qquad \sf  \dashrightarrow \: 5 \sqrt{2 }\:  \: units

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