How do you find the possible values for a if the points (5,8), (a,2) has a distance of 35?

1 Answer
Apr 16, 2017

Use the distance formula and solve for a:
{a:a=2,8}

Explanation:

The distance between the two points (x1,y1) and (x2,y2) is given as d=(x2x1)2+(y2y1)2.

For this problem, it is given that (x1,y1)=(5,8), (x2,y2)=(a,2), and d=35. We need to solve for a by plugging these values into the distance formula.
35=(a5)2+(28)2.

To solve for a, begin by squaring both sides of the equation and simplify a bit.
(35)2=(a5)2+(28)22
3252=(a5)2+(28)22
95=(a5)2+(28)2
45=(a5)2+(-6)2
45=a210a+25+36
45=a210a+61

Now subtract 45 from both sides to get an equation in quadratic form.
4545=a210a+6145
0=a210a+16

Finally, solve for a using either the quadratic formula or the factoring method. I'll be factoring.
0=a22a8a+16
0=a(a2)8(a2)
0=(a8)(a2)
a=8or2

Therefore the possible values for a are defined by the set {a:a=2,8}