Question

Find all points having an \(x\) -coordinate of 3 whose distance from the point (-2,-1) is \(13 .\) (a) By using the Pythagorean Theorem. (b) By using the distance formula.

Short answer

The points are (3, 11) and (3, -13).

Step-by-step solution

Understanding the problem

The task is to find all points (x, y) with an x-coordinate of 3 that are 13 units away from the point (-2, -1).

Using the given x-coordinate

Since the x-coordinate is fixed at 3, the coordinates of the point can be denoted as (3, y).

Applying the Pythagorean Theorem

Using the Pythagorean theorem in the context of distance, where the point (3, y) and (-2, -1) form a right triangle with the distance as the hypotenuse: \[(x_2 - x_1)^2 + (y_2 - y_1)^2 = distance^2\] Substituting the known points and distance: \[(3 - (-2))^2 + (y - (-1))^2 = 13^2\]

Simplifying the equation

Calculate the difference in x-coordinates: \[(3 + 2)^2 + (y + 1)^2 = 13^2\]Further simplifying gives: \[5^2 + (y + 1)^2 = 169\] \[25 + (y + 1)^2 = 169\]

Solving for y

Isolate \((y + 1)^2\): \[(y + 1)^2 = 144\]Take the square root of both sides: \[y + 1 = \pm 12\]Solve the two possible equations: \[y + 1 = 12\] \[y = 11\]and \[y + 1 = -12\] \[y = -13\]

Conclusion of part (a)

The points are (3, 11) and (3, -13).

Using the Distance Formula

Apply the distance formula directly. The distance formula for two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = distance\]

Setting up distance formula

Substitute the known points and distance: \[sqrt{(3 - (-2))^2 + (y - (-1))^2} = 13\]Simplify inside the square root: \[sqrt{5^2 + (y + 1)^2} = 13\]This simplifies to: \[sqrt{25 + (y + 1)^2} = 13\]

Removing the square root

Square both sides to eliminate the square root: \[25 + (y + 1)^2 = 169\]

Solving for y

Isolate \((y + 1)^2\): \[(y + 1)^2 = 144\]Take the square root of both sides: \[y + 1 = \pm 12\]Solve the two possible equations: \[y + 1 = 12\] \[y = 11\]and \[y + 1 = -12\] \[y = -13\]

Conclusion of part (b)

The points are (3, 11) and (3, -13).

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This can be expressed as: \[a^2 + b^2 = c^2\]
To use it in coordinate geometry to find the distance between two points, consider these steps:

• Identify the coordinates of the two points.
• Calculate the differences in the x-coordinates and y-coordinates.
• Apply the Pythagorean Theorem by squaring these differences and adding them together.

For example, given points \((3, y)\) and \((-2, -1)\):
\[(-2 - 3)^2 + (-1 - y)^2 = 13^2\]
Simplify to find the distance.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometry using a coordinate system. This often involves the use of the Cartesian coordinate system, where points are defined by pairs (or triples) of numbers.
To solve problems using coordinate geometry, follow these steps:

• Plot the points on a coordinate grid if it helps.
• Use algebraic methods to manipulate the coordinates.
• Apply relevant geometric formulas, such as the distance formula or the midpoint formula.

In this exercise, we are using the coordinate values and combining them with algebra to find the distances and solve for unknowns.

Solving Equations

To find unknown values, solving equations is a crucial skill. Here are the basic steps:

• Identify the equation you need to solve.
• Isolate the variable by using algebraic operations such as addition, subtraction, multiplication, and division.
• If necessary, use roots or exponents to solve.

In this exercise, we solve for \(y\) after isolating \((y + 1)^2\):
\[(y + 1)^2 = 144\]
Taking the square root on both sides:
\[ y + 1 = \pm 12\]
This leads us to the two solutions:
\[ y = 11\] and \[y = -13\]