1-3 Additional Practice Midpoint And Distance

1-3 Additional Practice: Midpoint and Distance Formulas

The midpoint and distance formulas are fundamental concepts in coordinate geometry, providing the tools to analyze segments and points on a Cartesian plane. Mastering these formulas is crucial for success in various mathematical applications, from simple geometry problems to advanced calculus concepts. This article provides comprehensive practice problems, detailed solutions, and insightful tips to help you solidify your understanding of these essential tools.

Section 1: Review of the Midpoint and Distance Formulas

Before diving into the practice problems, let's briefly review the formulas themselves. Understanding the underlying principles is key to effectively solving problems.

1. The Midpoint Formula:

The midpoint formula calculates the coordinates of the point exactly halfway between two given points. If we have two points, A(x₁, y₁) and B(x₂, y₂), the midpoint M(xₘ, yₘ) is given by:

xₘ = (x₁ + x₂) / 2

yₘ = (y₁ + y₂) / 2

Essentially, you're finding the average of the x-coordinates and the average of the y-coordinates.

2. The Distance Formula:

The distance formula calculates the straight-line distance between two points on a Cartesian plane. Using the same points A(x₁, y₁) and B(x₂, y₂), the distance 'd' is given by:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

This formula is a direct application of the Pythagorean theorem, where the distance is the hypotenuse of a right-angled triangle formed by the difference in x-coordinates and the difference in y-coordinates.

Section 2: Practice Problems – Midpoint Formula

Let's tackle some practice problems focusing on the midpoint formula. Remember to show your work – understanding the process is as important as arriving at the correct answer.

Problem 1: Find the midpoint of the line segment connecting the points A(-2, 5) and B(4, -3).

Solution:

1. Identify coordinates: x₁ = -2, y₁ = 5, x₂ = 4, y₂ = -3

2. Apply the midpoint formula:

   xₘ = (-2 + 4) / 2 = 1

   yₘ = (5 + (-3)) / 2 = 1

3. State the midpoint: The midpoint is M(1, 1).

Problem 2: The midpoint of a line segment is (3, -1). One endpoint is (1, 2). Find the coordinates of the other endpoint.

Solution:

1. Let the unknown endpoint be (x, y). We know the midpoint (3, -1) and one endpoint (1, 2).

2. Apply the midpoint formula:

   3 = (1 + x) / 2 => 6 = 1 + x => x = 5

   -1 = (2 + y) / 2 => -2 = 2 + y => y = -4

3. State the coordinates of the other endpoint: The other endpoint is (5, -4).

Problem 3: Points A, B, and C are collinear. If A = (-1, 4) and the midpoint of AB is M = (2, 1), find the coordinates of B. Then, if the midpoint of BC is N = (5, -2), find the coordinates of C.

Solution:

• Finding B:

  1. Use the midpoint formula with A(-1, 4) and M(2,1) to find B(x, y):

     2 = (-1 + x) / 2 => x = 5

     1 = (4 + y) / 2 => y = -2

  2. Therefore, B = (5, -2).

• Finding C:

  1. Use the midpoint formula with B(5, -2) and N(5, -2) to find C(x, y):

     5 = (5 + x) / 2 => x = 5

     -2 = (-2 + y) / 2 => y = -2

  2. Therefore, C = (5,-2). Notice that B and C are the same point, indicating that the points are collinear and likely represent a degenerate case (all points lie on the same location).

Section 3: Practice Problems – Distance Formula

Now let's move on to practice problems utilizing the distance formula. Remember to simplify your answers as much as possible.

Problem 4: Find the distance between the points P(1, 2) and Q(4, 6).

Solution:

1. Identify coordinates: x₁ = 1, y₁ = 2, x₂ = 4, y₂ = 6

2. Apply the distance formula:

   d = √[(4 - 1)² + (6 - 2)²] = √[3² + 4²] = √(9 + 16) = √25 = 5

3. State the distance: The distance between P and Q is 5 units.

Problem 5: Determine if the triangle with vertices A(1, 1), B(4, 5), and C(1, 5) is a right-angled triangle.

Solution:

1. Calculate the lengths of the sides using the distance formula:

   • AB = √[(4 - 1)² + (5 - 1)²] = √(9 + 16) = 5
   • BC = √[(1 - 4)² + (5 - 5)²] = √9 = 3
   • AC = √[(1 - 1)² + (5 - 1)²] = √16 = 4

2. Check for Pythagorean triples: Since 3² + 4² = 5², the Pythagorean theorem holds true.

3. Conclusion: Triangle ABC is a right-angled triangle.

Problem 6: A circle has its center at (2, -3) and passes through the point (5, 1). Find the radius of the circle.

Solution:

1. The radius is the distance between the center and any point on the circle.

2. Apply the distance formula: The center is (2, -3) and the point on the circle is (5, 1).

   radius = √[(5 - 2)² + (1 - (-3))²] = √(3² + 4²) = √25 = 5

3. State the radius: The radius of the circle is 5 units.

Problem 7: Find the equation of a circle with center (-1, 2) and passing through (3, 4).

Solution:

1. The radius is the distance between the center (-1, 2) and the point (3, 4).

   radius = √[(3 - (-1))² + (4 - 2)²] = √(16 + 4) = √20

2. The equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

3. Substitute the values: (x + 1)² + (y - 2)² = 20

Section 4: Advanced Practice Problems

These problems combine both the midpoint and distance formulas, requiring a deeper understanding of their applications.

Problem 8: The endpoints of the diameter of a circle are A(-2, 3) and B(4, -1). Find the equation of the circle.

Solution:

1. Find the midpoint (center of the circle):

   xₘ = (-2 + 4) / 2 = 1

   yₘ = (3 + (-1)) / 2 = 1

   Center = (1, 1)

2. Find the radius (half the diameter): Use the distance formula between the center (1, 1) and one endpoint, say A(-2, 3):

   radius = √[(-2 - 1)² + (3 - 1)²] = √(9 + 4) = √13

3. State the equation of the circle: (x - 1)² + (y - 1)² = 13

Problem 9: Points A(1, 2), B(4, y), and C(7, 6) are collinear. Find the value of y.

Solution:

1. Use the fact that the midpoint of AB is equal to the midpoint of BC (or AC, depending on the order of points). Let's use the midpoint of AB and the midpoint of BC.

2. Midpoint of AB: ((1+4)/2, (2+y)/2) = (5/2, (2+y)/2)

3. Midpoint of BC: ((4+7)/2, (y+6)/2) = (11/2, (y+6)/2)

4. Since the points are collinear, the midpoints must be collinear too. This might not hold true if the points are not in the correct order. We can use the concept of slope. The slope of AB must equal the slope of BC.

   Slope AB = (y-2)/(4-1) = (y-2)/3 Slope BC = (6-y)/(7-4) = (6-y)/3

   Setting them equal: (y-2)/3 = (6-y)/3 y-2 = 6-y 2y = 8 y = 4

5. Therefore, the value of y is 4.

Section 5: Tips for Success

• Draw diagrams: Visualizing the problem on a Cartesian plane can significantly aid understanding and problem-solving.

• Label your work: Clearly label variables and steps to avoid confusion.

• Check your answers: Substitute your answers back into the original problem to verify their correctness.

• Practice regularly: Consistent practice is crucial for mastering these formulas and developing problem-solving skills.

By working through these practice problems and understanding the underlying concepts, you'll build a strong foundation in coordinate geometry and be well-prepared for more advanced mathematical challenges. Remember, the key to success is consistent practice and a thorough understanding of the formulas. Keep practicing, and you'll become proficient in using the midpoint and distance formulas!