Solve each problem. To visualize the situation, use graph...

Solve each problem. To visualize the situation, use graph paper and a pair of compasses to carefully draw the graphs of the circles. Suppose that receiving stations $X, Y,$ and $Z$ are located on a coordinate plane at the points $(7,4),(-9,-4),$ and $(-3,9),$ respectively. The epicenter of an earthquake is determined to be 5 units from $X, 13$ units from $Y$, and 10 units from $Z$. Where on the coordinate plane is the epicenter located?

Solve each problem. To visualize the situation, use graph paper and a pair of compasses to carefully draw the graphs of the circles.
Suppose that receiving stations $X, Y,$ and $Z$ are located on a coordinate plane at the points $(7,4),(-9,-4),$ and $(-3,9),$ respectively. The epicenter of an earthquake is determined to be 5 units from $X, 13$ units from $Y$, and 10 units from $Z$. Where on the coordinate plane is the epicenter located?

Key Concepts

Distance Formula

The distance formula in coordinate geometry is used to determine the distance between any two points in the plane. It derives from the Pythagorean theorem and is fundamental when comparing distances, such as the fixed distances from the epicenter to given points. This formula underpins the formulation of the circle equations based on radii and centers.

Circle Equation

A circle equation represents all the points in the plane at a fixed distance (the radius) from a central point. In the context of the problem, each receiving station and its associated distance from the epicenter form a circle. Writing the circle equations allows for the establishment of relationships between coordinates that must be satisfied simultaneously.

System of Equations

When multiple circle equations are expressed algebraically, solving for their intersection point entails setting up a system of equations. This process involves algebraic methods such as substitution or elimination to accurately determine the coordinates that satisfy all the conditions, which in this problem represents the earthquake epicenter.

Graphical Representation

Graphing the circles using tools like graph paper and compasses offers a visual approach to understanding the problem. This method aids in estimating the location of the intersection (the epicenter) and provides insight into the nature of the solution, such as whether one, two, or no intersection points exist.

Transcript

00:03 Okay, let's visualize this situation.

00:07 So i've already plotted points x, y, and z.

00:10 And we know that the epicenter of the earthquake was five units from x.

00:15 So that means it falls along a circle that's five units from x.

00:19 So if i picture that circle, i'm going to go out five units as my radius and then just kind of roughly sketch something like this.

00:30 Okay? and then same idea for y.

00:34 We know the epicenter was 13 units from point y.

00:38 So then if we draw a circle that has a radius of 13 and a center at point y, it's going to be roughly something like this.

00:52 Whoops, maybe not quite that.

00:55 Okay, that's part of the circle.

00:58 And then for z, the epicenter was 10 units from point z.

01:02 So we would have a circle with center z and a radius of 10.

01:06 So if we go 10 units, that's going to be something like that.

01:13 Okay? so our goal is to find that epicenter, and it looks like from our sketch that maybe just maybe, there's a point of intersection of all three of those circles right about there.

01:26 So our goal is going to be to find that point of intersection.

01:30 So suppose i write the equations of the circles.

01:33 So for the red circle, it would be x minus 7 quantity squared.

01:38 Plus y minus 4 quantity squared equals 25 and for the blue circle it would be x plus 9 quantity squared plus y plus 4 quantity squared equals 169 and then for the green circle we would have x plus 3 quantity squared plus y minus 9 quantity squared is equal to 100 okay so we're looking for the point point of intersection of these three.

02:17 Okay, so what we're trying to do then is find the point of intersection of these three curves.

02:22 So there's a lot of algebra ahead.

02:25 And what i'm going to do next is i'm going to multiply each equation out so that it doesn't have parentheses anymore.

02:31 So the first one is going to be x squared minus 14x plus 49 plus y squared minus 8y plus 16 equals 25.

02:43 The second one is going to be x squared plus 18x plus 81 plus y squared plus 8y plus 16 equals 169.

02:58 And then treating this like a system of equations, we can go ahead and subtract these two equations, and that's going to cancel the x squared terms and the y squared terms.

03:08 And that's going to give us negative 32x minus 32.

03:14 Minus 18y equals negative 144.

03:21 Let's multiply both sides of this by negative 1...

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