Right Triangle PQR: Find X,y Relationship And X Values

by TextBrain Team 55 views

Hey guys! Let's dive into a cool math problem involving right triangles and coordinate geometry. We've got a triangle PQR where the vertices are P(x, y), Q(-2, -3), and R(2, 3). The coolest part? This triangle is right-angled at P! Our mission, should we choose to accept it (and we do!), is to figure out the relationship between x and y and then find all possible values of x when y is 2. Sounds like fun? Let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what's going on. We have three points forming a triangle, and one of the angles is a right angle. This is super important because it allows us to use some awesome properties of right triangles, like the Pythagorean theorem and the concept of perpendicular lines. We need to find a connection between the coordinates of point P (x and y) based on the fact that angle P is 90 degrees. Once we have that relationship, we can plug in y = 2 and solve for x. Think of it as a mathematical treasure hunt!

Key Concepts

To solve this problem effectively, we'll be using a few key concepts from coordinate geometry:

  • Distance Formula: This helps us calculate the distance between two points in a coordinate plane. Remember, the distance d between points (x1, y1) and (x2, y2) is given by: d = √((x2 - x1)² + (y2 - y1)²)
  • Slope of a Line: The slope tells us how steep a line is. The slope m of a line passing through points (x1, y1) and (x2, y2) is given by: m = (y2 - y1) / (x2 - x1)
  • Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1. This is the golden nugget we'll use because we know angle P is a right angle!

With these concepts in our toolkit, we're ready to tackle the problem. Let's roll!

Finding the Relationship Between x and y

Our main goal here is to establish the relationship between the x and y coordinates of point P. Remember, the triangle PQR is right-angled at P. This means that the lines PQ and PR are perpendicular to each other. And as we just discussed, perpendicular lines have slopes whose product is -1. This is our key to unlocking the relationship!

Calculating the Slopes

First, let's find the slopes of the lines PQ and PR. We'll use the slope formula we talked about earlier:

  • Slope of PQ (mPQ): Using points P(x, y) and Q(-2, -3), we have:

    mPQ = (-3 - y) / (-2 - x)

  • Slope of PR (mPR): Using points P(x, y) and R(2, 3), we have:

    mPR = (3 - y) / (2 - x)

Now we have expressions for the slopes of PQ and PR in terms of x and y. The next step is to use the perpendicularity condition.

Applying the Perpendicularity Condition

Since PQ and PR are perpendicular, the product of their slopes must be -1. So, we can write:

mPQ * mPR = -1

Substituting the expressions we found for the slopes, we get:

[(-3 - y) / (-2 - x)] * [(3 - y) / (2 - x)] = -1

This looks a bit intimidating, but don't worry, we'll simplify it. Let's multiply both sides by (-2 - x)(2 - x) to get rid of the fractions:

(-3 - y)(3 - y) = -1 * (-2 - x)(2 - x)

Simplifying the Equation

Now, let's expand the products on both sides of the equation:

(-9 + 3y - 3y + y²) = -1 * (-4 + x²)

Notice that the +3y and -3y terms cancel out on the left side. This simplifies things nicely:

y² - 9 = 4 - x²

Now, let's rearrange the equation to get all the variables on one side:

x² + y² = 13

Boom! We've done it! This is the relationship between x and y that we were looking for. It's a beautiful equation that tells us how the coordinates of point P must be related for triangle PQR to be right-angled at P.

Finding Possible Values of x When y = 2

Now that we have the relationship between x and y (x² + y² = 13), the second part of our mission is much easier. We're given that y = 2, and we need to find all possible values of x that satisfy this condition.

Substituting y = 2 into the Equation

Let's plug y = 2 into our equation:

x² + (2)² = 13

This simplifies to:

x² + 4 = 13

Solving for x

Now, let's isolate x²:

x² = 13 - 4

x² = 9

To find the values of x, we need to take the square root of both sides:

x = ±√9

So, we have two possible values for x:

x = 3 or x = -3

The Possible Values of x

Therefore, when y = 2, the possible values of x are 3 and -3. This means there are two possible locations for point P that would make triangle PQR a right-angled triangle at P: (3, 2) and (-3, 2).

Conclusion: Mission Accomplished!

Guys, we did it! We successfully found the relationship between x and y for point P in a right-angled triangle PQR, and then we used that relationship to find the possible values of x when y is 2. We used key concepts like the distance formula, slope of a line, and the crucial property of perpendicular lines having slopes that multiply to -1.

Recap of the Solution

  1. We started by understanding the problem and identifying the key concepts we would need.
  2. We calculated the slopes of lines PQ and PR in terms of x and y.
  3. We applied the perpendicularity condition (mPQ * mPR = -1) and simplified the equation to find the relationship: x² + y² = 13.
  4. We substituted y = 2 into the relationship and solved for x, finding two possible values: x = 3 and x = -3.

This problem demonstrates how different concepts in coordinate geometry can come together to solve a challenging problem. Keep practicing, and you'll become a math whiz in no time! Keep your mind sharp, and math will become your superpower!