Chapter 1.8, Problem 33E

To determine

To calculate: The closer point among $A\left(6,7\right)$ and $B\left(-5,8\right)$ to the origin.

Answer to Problem 33E

Thecloser point to originis $A\left(6,7\right)$ .

Explanation of Solution

Given information:

The points $A\left(6,7\right)$ and $B\left(-5,8\right)$ .

Formula used:

Distance $d\left(A,B\right)$ between two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Calculation:

Consider the provided set of points $A\left(6,7\right)$ and $B\left(-5,8\right)$ .

Recall that the distance $d\left(A,B\right)$ between two points $A=\left(x_1,y_1\right)\text{ and }B=\left(x_2,y_2\right)$ in the Cartesian plane is denoted by $d\left(A,B\right)=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$ .

Denote the origin as $O\left(0,0\right)$ .

Evaluate the distance between $A\left(6,7\right)$ and $O\left(0,0\right)$ .

  $\begin{array}{c}d\left(A,O\right)=\sqrt{{\left(0-6\right)}^2+{\left(0-7\right)}^2}\\ =\sqrt{36+49}\\ =\sqrt{85}\end{array}$

Next evaluate the distance between $B\left(-5,8\right)$ and $O\left(0,0\right)$ .

  $\begin{array}{c}d\left(B,O\right)=\sqrt{{\left(0-\left(-5\right)\right)}^2+{\left(0-8\right)}^2}\\ =\sqrt{25+64}\\ =\sqrt{89}\end{array}$

Observe that $d\left(B,O\right)>d\left(A,O\right)$ .

That is the distance between the point $A\left(6,7\right)$ and the origin is less than the distance the point $B\left(-5,8\right)$ and the origin.

Therefore, the point $A\left(6,7\right)$ is near to origin.

Thus, the closer point to origin is $A\left(6,7\right)$ .