Q. 10CE - Geometry

Question

10. Find the distance between the points named. Give all answers in simplest

form.

1.  (0, 0) and (5, -3).
2.  (3, -2) and (-5, -2).

      3. (4, 4) and (-3, -3).

Step-by-Step Solution

Answer

1. The distance is $\sqrt{34}$.
2. The distance is 8.
3. The distance is $\sqrt{98}$

a. Step-1 – Given

The given points are (0,0) and (5, -3). 

Step-2 – To determine

We have to find the distance between the points.

Step-3 – Calculation

We will use the distance formula.

$d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

with $x_1=0,x_2=5,y_1=0\text{ and }{y}_2=-3$

$\begin{array}{l}d=\sqrt{{\left(5-0\right)}^2+{\left(-3-0\right)}^2}\left[d=\text{distance}\right]\\ d=\sqrt{{\left(5\right)}^2+{\left(-3\right)}^2}\\ d=\sqrt{25+9}\\ d=\sqrt{34}\end{array}$

So, the distance is$\sqrt{34}$ .

b. Step-1 – Given

The given points are (3, -2) and (-5, -2). 

Step-2 – To determine

We have to find the distance between the points.

Step-3 – Calculation

We will use the distance formula.

$d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

With $x_1=3,x_2=-5,y_1=-2\text{ and }{y}_2=-2$

$\begin{array}{l}d=\sqrt{{\left(-5-3\right)}^2+{\left(-2-\left(-2\right)\right)}^2}\left[d=\text{distance}\right]\\ d=\sqrt{{\left(-8\right)}^2+{\left(-2+2\right)}^2}\\ d=\sqrt{{\left(-8\right)}^2+{\left(0\right)}^2}\\ d=\sqrt{64+0}\\ d=\sqrt{64}\\ d=8\end{array}$

So, the distance is 8.

c. Step-1 – Given

The given points are (4, 4) and (-3, -3). 

Step-2 – To determine

We have to find the distance between the points.

Step-3 – Calculation

We will use the distance formula.

$d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

with $x_1=4,x_2=-3,y_1=4\text{ and }{y}_2=-3$

$\begin{array}{l}d=\sqrt{{\left(-3-4\right)}^2+{\left(-3-4\right)}^2}\left[d=\text{distance}\right]\\ d=\sqrt{{\left(-7\right)}^2+{\left(-7\right)}^2}\\ d=\sqrt{49+49}\\ d=\sqrt{98}\end{array}$

So, the distance is$\sqrt{98}$ .