Solving Coordinate Geometry Problems Distance And Triangle Identification

by ADMIN 74 views

#h1 Introduction

In coordinate geometry, understanding the relationships between points, distances, and geometric shapes is fundamental. This article delves into two problems that highlight these concepts. The first problem involves calculating the value of a coordinate given a distance between two points, while the second focuses on identifying the type of triangle formed by three given points. Through detailed explanations and step-by-step solutions, we aim to provide a comprehensive understanding of these concepts.

#h2 Problem 1 Finding the Value of p Given Distance

Understanding the Distance Formula

At the heart of this problem lies the distance formula, a crucial tool in coordinate geometry. The distance formula is derived from the Pythagorean theorem and allows us to calculate the distance between two points in a coordinate plane. If we have two points, (x1, y1) and (x2, y2), the distance d between them is given by:

d = √((x2 - x1)² + (y2 - y1)²)

This formula essentially calculates the length of the hypotenuse of a right triangle formed by the difference in the x-coordinates and the difference in the y-coordinates. Understanding the distance formula is paramount for solving a wide range of problems in coordinate geometry, from finding the lengths of line segments to determining the properties of geometric shapes.

Applying the Distance Formula to the Problem

In our problem, we are given two points, (4, p) and (1, 0), and the distance between them, which is 5 units. Our goal is to find the value(s) of p. To do this, we will apply the distance formula and set up an equation:

5 = √((1 - 4)² + (0 - p)²)

This equation represents the relationship between the given points and the distance. The next step is to solve for p. We begin by squaring both sides of the equation to eliminate the square root:

25 = (1 - 4)² + (0 - p)²
25 = (-3)² + (-p)²
25 = 9 + p²

Now, we isolate p²:

p² = 25 - 9
p² = 16

To find the values of p, we take the square root of both sides:

p = ±√16
p = ±4

Therefore, the possible values of p are 4 and -4. This means there are two points, (4, 4) and (4, -4), that are 5 units away from the point (1, 0).

Interpreting the Results

The fact that we obtained two values for p indicates that there are two possible locations for the point (4, p) that satisfy the given condition. This highlights an important aspect of coordinate geometry: a single distance constraint can sometimes lead to multiple solutions. In this case, the points (4, 4) and (4, -4) are reflections of each other across the x-axis, both lying on a circle centered at (1, 0) with a radius of 5 units. The two values, p = 4 and p = -4, represent the y-coordinates of these two points, demonstrating the symmetry inherent in the problem.

Conclusion for Problem 1

In conclusion, by applying the distance formula and solving the resulting equation, we found that the possible values for p are 4 and -4. This exercise demonstrates the power of the distance formula in solving geometric problems and highlights the importance of considering all possible solutions. Understanding the interplay between algebraic equations and geometric concepts is crucial for success in coordinate geometry.

#h2 Problem 2 Identifying the Type of Triangle

Understanding Triangle Properties

Before we tackle the second problem, it's crucial to review the properties of different types of triangles. Triangles can be classified based on their side lengths and angles. The main types of triangles we'll consider are:

  • Equilateral triangle: A triangle with all three sides of equal length and all three angles equal to 60 degrees.
  • Isosceles triangle: A triangle with two sides of equal length and two equal angles.
  • Right-angled triangle: A triangle with one angle equal to 90 degrees. This triangle satisfies the Pythagorean theorem: a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.
  • Scalene triangle: A triangle with all three sides of different lengths and all three angles of different measures.

Knowing these definitions is essential for correctly identifying the type of triangle formed by the given points. Our approach will involve calculating the lengths of the sides and then applying the appropriate criteria to classify the triangle. A solid understanding of triangle properties and how they relate to side lengths and angles is essential for solving this problem. The relationships between side lengths, such as the Pythagorean theorem for right-angled triangles or the equal sides in isosceles and equilateral triangles, are key to triangle classification.

Calculating Side Lengths

To determine the type of triangle formed by the points (2, 5), (4, -1), and (6, -7), we first need to calculate the lengths of its sides. We'll use the distance formula again for this purpose. Let's label the points as A(2, 5), B(4, -1), and C(6, -7). We will now calculate the lengths of sides AB, BC, and CA.

Length of AB

Using the distance formula between points A(2, 5) and B(4, -1):

AB = √((4 - 2)² + (-1 - 5)²)
AB = √((2)² + (-6)²)
AB = √(4 + 36)
AB = √40

Length of BC

Using the distance formula between points B(4, -1) and C(6, -7):

BC = √((6 - 4)² + (-7 - (-1))²)
BC = √((2)² + (-6)²)
BC = √(4 + 36)
BC = √40

Length of CA

Using the distance formula between points C(6, -7) and A(2, 5):

CA = √((2 - 6)² + (5 - (-7))²)
CA = √((-4)² + (12)²)
CA = √(16 + 144)
CA = √160

Now we have the lengths of the three sides: AB = √40, BC = √40, and CA = √160. These side lengths will help us classify the triangle.

Classifying the Triangle

Having calculated the side lengths, we can now determine the type of triangle. We have AB = √40, BC = √40, and CA = √160. Notice that two sides, AB and BC, have equal lengths. This immediately tells us that the triangle is an isosceles triangle, as it has two sides of equal length. However, to further classify the triangle, we need to check if it is also a right-angled triangle.

To check for a right angle, we will use the Pythagorean theorem. If the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides, then the triangle is right-angled. In our case, CA is the longest side, so we need to check if CA² = AB² + BC².

CA² = (√160)² = 160
AB² = (√40)² = 40
BC² = (√40)² = 40

Now, let's see if the Pythagorean theorem holds:

AB² + BC² = 40 + 40 = 80

Since CA² (160) is not equal to AB² + BC² (80), the triangle is not a right-angled triangle. Therefore, the triangle is an isosceles triangle but not a right-angled triangle.

Conclusion for Problem 2

By calculating the lengths of the sides using the distance formula and then applying the properties of triangles, we determined that the triangle formed by the points (2, 5), (4, -1), and (6, -7) is an isosceles triangle. This problem reinforces the importance of using the distance formula to find side lengths and understanding the characteristics of different types of triangles for classification. The combination of these skills is essential for solving geometry problems in the coordinate plane.

#h2 Summary and Key Takeaways

In this article, we tackled two problems in coordinate geometry that demonstrate the application of the distance formula and the properties of triangles. In the first problem, we found the value of p given the distance between two points, which required us to apply the distance formula and solve a quadratic equation. We learned that a single distance constraint can have multiple solutions, highlighting the importance of considering all possibilities.

The second problem involved identifying the type of triangle formed by three points. We calculated the lengths of the sides using the distance formula and then used the side lengths to classify the triangle. We found that the triangle was isosceles because two of its sides were equal in length. We also checked if it was a right-angled triangle using the Pythagorean theorem, but it didn't satisfy the condition. This problem emphasized the significance of understanding the characteristics of different types of triangles and applying the appropriate criteria for classification.

These exercises illustrate the fundamental concepts of coordinate geometry and how they can be applied to solve practical problems. The distance formula is a cornerstone of coordinate geometry, and a solid understanding of triangle properties is crucial for geometric problem-solving. By mastering these concepts, one can confidently tackle a wide range of problems in this field.

#h2 Final Thoughts

Coordinate geometry provides a powerful framework for analyzing geometric shapes and figures using algebraic techniques. The problems we've explored here are just a glimpse of the rich landscape of this field. By combining geometric intuition with algebraic tools, we can gain a deeper understanding of the relationships between points, lines, and shapes in the coordinate plane. The distance formula and triangle properties are essential components of this framework, and their mastery opens the door to solving more complex and fascinating problems in geometry.