Of the following six points, four are an equal distance from...

Of the following six points, four are an equal distance from the point $A(2,3)$ and two are not. (a) Identify which four, and (b) find any two additional points that are this same (nonvertical, nonhorizontal) distance from ( 2,3 ): $B(7,15) \quad C(-10,8) \quad D(9,14) \quad E(-3,-9)$ $F(5,4+3 \sqrt{10}) \quad G(2-2 \sqrt{30}, 10)$

Of the following six points, four are an equal distance from the point $A(2,3)$ and two are not.
(a) Identify which four, and (b) find any two additional points that are this same (nonvertical, nonhorizontal) distance from ( 2,3 ): $B(7,15) \quad C(-10,8) \quad D(9,14) \quad E(-3,-9)$ $F(5,4+3 \sqrt{10}) \quad G(2-2 \sqrt{30}, 10)$

Key Concepts

Distance Formula

The distance formula, derived from the Pythagorean theorem, allows one to compute the distance between two points in the Euclidean plane by using their coordinates. It is a fundamental tool in coordinate geometry to determine how far apart points are and is expressed as the square root of the sum of the squared differences in each coordinate.

Circle in Coordinate Geometry

A circle is defined as the set of all points that are a fixed distance, called the radius, from a given point, known as the center. In coordinate geometry, this concept is crucial when working with problems involving equidistant points, as any point that maintains the same distance from a specified center lies on a circle.

Equidistance Analysis

Analyzing which points are equidistant from a given point requires applying the distance formula to ensure that selected points share the same distance, effectively lying on the same circle. This concept is essential for solving problems where filtering out or identifying groups of points based on their distance from a reference point is necessary.

Transcript

00:01 This question consists of two parts.

00:03 We will start with part a.

00:05 First we will find the distance between point a and b.

00:13 Point a is 2 .3 and point b is 7 .15.

00:23 We will apply distance formula here under root 7 minus 2 whole square plus 15 minus 3 whole square.

00:34 On solving this we get the value 13.

00:39 Now we will find distance between point a and c.

00:44 Point a is 2 .3 and point c is minus 10 comma 8 now we will apply distance formula under root minus 10 minus 2 whole square plus 8 minus 3 whole square on solving this we get the value 13 now we will find distance between point a and d point a is 2 .3 and point d and point d is 9 .14 we will apply distance formula here under under root 9 minus 2 whole square plus 14 minus 3 whole square.

01:39 On solving this, we get the value under root 170.

01:49 Now we will find distance between point a and e.

01:54 Point a is 2 .3.

01:57 Point e is minus 3.

02:00 Minus 9.

02:01 We will apply a distance formula here under root minus 3 minus 2 whole square plus minus 9 minus 3 whole square.

02:15 On solving this, we get the value 13.

02:21 Now we will find distance between point a and f.

02:26 Point a is 2 .3 and point f is 5 .4 plus 3 under root 10.

02:37 We will apply a distance formulae under root 5 minus 2 whole square plus 4 plus 3 under root 10 minus 3 whole square.

02:54 On solving this, we get the value under root 40 plus 6 under root 10.

03:09 Now we will find the distance between point a and g...