Question

find the distance d(a,b) between the points a and b. a(\frac{\sqrt{57}}{2},\frac{\sqrt{43}}{2}); b(0,0) d(a,b)= (simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}$. Here, $x_1=\frac{\sqrt{57}}{2}$, $y_1=\frac{\sqrt{43}}{2}$, $x_2 = 0$, $y_2=0$.

Step2: Substitute values

$d(A,B)=\sqrt{(\frac{\sqrt{57}}{2}-0)^2+(\frac{\sqrt{43}}{2}-0)^2}=\sqrt{(\frac{\sqrt{57}}{2})^2+(\frac{\sqrt{43}}{2})^2}$.

Step3: Simplify squares

$(\frac{\sqrt{57}}{2})^2=\frac{57}{4}$ and $(\frac{\sqrt{43}}{2})^2=\frac{43}{4}$. So $d(A,B)=\sqrt{\frac{57}{4}+\frac{43}{4}}$.

Step4: Add fractions

$\frac{57}{4}+\frac{43}{4}=\frac{57 + 43}{4}=\frac{100}{4}=25$. Then $d(A,B)=\sqrt{25}$.

Step5: Calculate square - root

$\sqrt{25}=5$.

Answer:

$5$