Notes on Geometric Sequence and Series

Tuesday, February 10, 2026

 Geometric Sequence and Series

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Key Answer to Rectangular Coordinate system

Thursday, May 15, 2025

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Plot Like a Pro: Master the Rectangular Coordinate System in Just 5 Steps!

 Understanding Rectangular Coordinate Systems Basics

1. Introduction 

• Imagine this: You're locked in an intense game of Battleship, heart pounding as you call out *"B-5!"* Your opponent hesitates... then groans—direct hit! 🎯 What’s your secret? You’re not just guessing—you’re using math to dominate the grid.

  It turns out that Battleship is really a stealthy lesson in the rectangular coordinate system—the same tool scientists use to map stars, engineers use to design cities, and even your phone uses to navigate. Master this, and you’ll not only crush your opponents in games but unlock the hidden math behind everything from GPS to video game design.

  Ready to turn coordinates into your superpower? Let’s dive in!

• Objective: By the end of this lesson, you will be able to:

  • Plot points accurately on the coordinate plane.

  • Identify the quadrant of a given point.

  • Calculate the distance between two points using the distance formula.

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2. Prerequisite Knowledge Check

Before diving in, make sure you understand:
✔ Number lines (positive and negative directions).
✔ Ordered pairs (e.g., (3, 4) means x=3, y=4).
✔ Basic operations (addition, subtraction, squaring).

Need a refresher?

• Number Lines Explained

• Ordered Pairs Practice

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3. Core Concept Explanation

What is the Rectangular Coordinate System?

• A grid formed by two perpendicular number lines:

  • x-axis (horizontal)

  • y-axis (vertical)

• Origin (0,0): Where the axes intersect.

• Quadrants: Four sections labeled I (+,+), II (−,+), III (−,−), IV (+,−).

Common Mistakes to Avoid:

❌ Swapping x and y coordinates (e.g., writing (y, x) instead of (x, y)).
❌ Forgetting negative signs when plotting points (e.g., (−2, 3) vs. (2, 3)).
❌ Misidentifying quadrants (e.g., (3,−5) is in Quadrant IV, not II).

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4. Worked Examples (Gradual Release of Responsibility)

Example 1 (I Do - Teacher Modeling)

Problem: Plot the point (2, 3).
Solution:

1. Start at the origin (0,0).

2. Move 2 units right along the x-axis (since x=2 is positive).

3. From there, move 3 units up along the y-axis (since y=3 is positive).

4. Mark the point where you land.
   Conclusion: (2, 3) is in Quadrant I.

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Example 2 (We Do - Guided Practice)

Problem: Plot the point (−1, 4).
Steps Together:

1. Start at (0,0).

2. Move 1 unit left (x=−1).

3. Move 4 units up (y=4).

4. Where is this point located?
   Answer: (−1, 4) is in Quadrant II.

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Example 3 (You Do Together - Collaborative Practice)

Problem: Identify the quadrant of (−3, −5).
Group Discussion:

• Is x negative or positive? (Negative)

• Is y negative or positive? (Negative)

• Which quadrant has (−,−)?
  Answer: Quadrant III.

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Example 4 (You Do - Independent Practice)

Problem: Calculate the distance between (1, 2) and (4, 6).
Formula: Distance = √((x₂−x₁)² + (y₂−y₁)²).
Worked Solution:

1. Subtract x-coordinates: 4−1 = 3 → 3² = 9.

2. Subtract y-coordinates: 6−2 = 4 → 4² = 16.

3. Add results: 9 + 16 = 25.

4. Take the square root: √25 = 5.
   Final Answer: The distance is 5 units.

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Example 5 (Challenge Problem - Extended Thinking)

Problem: Find the distance between (−2, 4) and (3, −1).
Steps:

1. Subtract x-coordinates: 3−(−2) = 5 → 5² = 25.

2. Subtract y-coordinates: −1−4 = −5 → (−5)² = 25.

3. Add results: 25 + 25 = 50.

4. Take the square root: √50 = 5√2.
   Final Answer: The distance is 5√2 units.

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5. Practice Problems (With Solutions)

Easy: Plot (0, −3). What quadrant is it located in?
Medium: What is the distance between (5, 1) and (2, −3)?
Hard: If point A is (−4, 0) and point B is (0, 3), what is the distance between them?

*Easy: (0,−3) lies on the **y-axis** (not in any quadrant). - 

**Medium: Distance = √((2−5)² + (−3−1)²) = √(9 + 16) = **5 units**. - 

***Hard: Distance = √((0−(−4))² + (3−0)²) = √(16 + 9) = **5 units**. 

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6. Real-World Applications

• GPS Navigation: Uses coordinates to pinpoint locations.

• Video Games: Characters move based on (x,y) positions.

• Architecture: Blueprints rely on grid systems for precision.

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7. Summary & Key Takeaways

✔ Points are written as (x, y).
✔ Quadrants are labeled counterclockwise: I → II → III → IV.
✔ Distance formula: √((x₂−x₁)² + (y₂−y₁)²).

 "X comes before Y in the alphabet, just like in (x,y)!"

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8. Further Reading & Resources

📺 Video: Coordinate Plane
📖 Book: Graphing Notebook
🔗 WOW Math: Math 8

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9. Interactive Element

Quick Quiz:

1. What quadrant is (−7, −2) in? (Answer: III)

2. What is the distance between (0,0) and (6,8)? (Answer: 10 units)

____________________________________________________________________________

10. Problem set. Fill in the blanks. Given the problems with a partial solution.

🔢 Problem Set: Coordinate Plane Puzzle Fixers

(For small groups of 3-4 students)*

Problem 1: The Misplaced Point

Partial Solution:
*"The point (3, -2) was plotted in Quadrant II, but that’s incorrect because..."*
Task:

1. Identify the error.

2. Correctly plot and label the point.

3. Challenge: Find a point in Quadrant II with the same y-coordinate.

Self-Check: The corrected point forms a rectangle with (-3, -2), (3, -2), and (-3, 2).

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Problem 2: Distance Detective

Partial Solution:
*"The distance between (1, 4) and (5, 1) is calculated as:
√‾(5−1)² + (1−4)² = √‾16 + ___ = √‾___ = 5 units."*
Task:

1. Fill in the missing steps.

2. Challenge: Find a point on the y-axis that’s exactly 5 units from (1, 4).

Hint: The missing term is a perfect square.

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Problem 3: Reflection Riddle

Partial Solution:
*"When (2, -3) is reflected over the x-axis, the new point is (2, 3). If you then reflect it over the y-axis, the final point is (___ , ___)."*
Task:

1. Complete the transformation.

2. Challenge: What single reflection would take (2, -3) directly to the final point?

Visual Aid: Provide small mirrors to test predictions.

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Problem 4: Battleship Logic

Partial Solution:
*"A ship stretches from (-1, 2) to (-1, 5). Its length is 3 units. If another ship runs from (3, -4) to (___, -4) and is twice as long, the missing x-coordinate is ___."*
Task:

1. Find the endpoint.

2. Challenge: Could both ships be sunk by hitting (-1, 4) and (5, -4)? Explain.

Self-Check: The completed ship covers 3 integer x-values.

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Problem 5: Midpoint Mystery

Partial Solution:
*"The midpoint between (-6, 3) and (2, -1) is:
( (-6 + ___)/2 , (3 + )/2 ) = ( , ___)."*
Task:

1. Fill in the blanks.

2. Challenge: Find the endpoint if (-2, 1) is the midpoint and the other endpoint is (0, 5).

Hint: The missing midpoint coordinates are integers.

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Problem 6: Shape 

Partial Solution:
"Points A(1, 1), B(1, 4), and C(5, 1) form a right triangle because the slopes of AB and AC are ___ and ___. The area is ___ square units."
Task:

1. Calculate the missing slopes and area.

2. Challenge: Add point D to make a rectangle.

Clue: Area = ½ × base × height.

📝 Practice Problems: Rectangular Coordinate System

Directions: Solve each problem with your group. Show all work and justify your answers.

Basic Skills (Problems 1-3)

1. Plotting Points
   Plot and label these points: A(2, 5), B(-3, 0), C(-1, -4). Identify the quadrant or axis for each.

2. Distance Warm-Up
   Find the distance between (0, 0) and (6, 8).

3. Missing Coordinate
   If (4, y) is 5 units from (1, 2), find *y*. (Hint: There are two possible answers!)

Mid-Level Challenges (Problems 4-6)

4. Quadrant Logic
   "Point P has a negative x-coordinate and a positive y-coordinate. If you reflect P over the y-axis, which quadrant is the new point in?"

5. Perimeter Puzzle
   Three vertices of a rectangle are at (1, 1), (1, 4), and (5, 4). Find:
   a) The fourth vertex.
   b) The perimeter.

6. Collinear Points
   Show whether (2, 3), (4, 6), and (0, 0) lie on the same straight line.

Advanced Applications (Problems 7-10)

7. Battleship Midpoint
   A ship’s midpoint is at (3, 2). One endpoint is (5, 5). Find the other endpoint.

8. Circle Equation
   A circle’s center is at (-1, 3) and passes through (2, 7). Find its radius.

9. Slope Analysis
   A line passes through (-2, 4) and (1, -2). Find:
   a) The slope.
   b) The y-intercept.

10. Real-World GPS
    You’re at (3, -1) and need to reach a café 10 units away at (x, 5). Find *x*.

Worksheet PDF Download

_________________________________________________________________________

🔑 Solutions

1. Plotting Points

   • A(2,5): Quadrant I

   • B(-3,0): On x-axis

   • C(-1,-4): Quadrant III

2. Distance
   √‾(6−0)² + (8−0)² = √‾36 + 64 = √‾100 = 10 units

3. Missing Coordinate
   √‾(4−1)² + (y−2)² = 5 → 9 + (y−2)² = 25 → *y* = 6 or -2

4. Reflection
   Original: Quadrant II → Reflected: Quadrant I

5. Rectangle
   a) Fourth vertex: (5, 1)
   b) Perimeter: 2(4 + 3) = 14 units

6. Collinear Check
   Slope (0,0)→(2,3) = 1.5; Slope (2,3)→(4,6) = 1.5 → Yes, collinear

7. Endpoint
   Midpoint formula → Other endpoint: (1, -1)

8. Radius
   √‾(2−(−1))² + (7−3)² = √‾9 + 16 = 5 units

9. Line Equation
   a) Slope = (−2−4)/(1−(−2)) = -2
   b) y = -2x + 0 → y-intercept: (0, 0)

10. GPS Café
    √‾(x−3)² + (5−(−1))² = 10 → (x−3)² + 36 = 100 → *x* = 11 or -5

_________________________________________________________________________

SIMPLIFIED TIPS FOR LEARNING.

🌟 5 Easy Tips to Master the Rectangular Coordinate System

1. Turn Coordinates into a Game 🎲

• Battleship Method: Play the classic game Battleship (or use graph paper) to practice plotting points like (3,5) and (-2,4).

• Treasure Hunts: Hide "treasure" in your house and describe its location using coordinates (e.g., "3 steps right from the door, 2 steps up").

2. Remember the "X to the Sky, Y to the Thigh" Trick ☝️

• X-axis: Think of it as the ground (left/right).

• Y-axis: Imagine it as a ladder (up/down).

• Quadrants: Use the sign rules:

  • Quadrant I (+, +): Happy (both positive)

  • Quadrant II (−, +): Sad left, happy up

  • Quadrant III (−, −): Sad all around

  • Quadrant IV (+, −): Happy right, sad down

3. Use Real-Life Examples 

• GPS: Google Maps uses coordinates to find locations.

• Sports: Basketball court positions (e.g., (0,0) = center court).

• Art: Pixel art and video game design rely on grids.

4. Master Formulas with Simple Stories 📖

• Distance Formula: Imagine a right triangle between two points. Use the Pythagorean Theorem:

  • "Walk the x-distance (base), climb the y-distance (height), then find the hypotenuse (distance)."
    Example: Distance between (1,2) and (4,6) = √((4-1)² + (6-2)²) = 5 units.

• Midpoint Formula: Think of it as averaging two locations:
  "Add the x’s, divide by 2. Add the y’s, divide by 2. Boom—middle point!"

5. Practice with Mini-Challenges ✏️

• Daily Drills: Plot 3 random points every day and name their quadrants.

• Flashcards: Write point pairs on one side and their distance/midpoint on the back.

• Error Analysis: Intentionally make mistakes (e.g., swap x/y) and correct them.

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🎯 Pro Tip:

Struggling with negatives? Use colored pens:

• 🔴 Red for negative coordinates

• 🔵 Blue for positive coordinates

Example: (-3, 5) = 3 red steps left, 5 blue steps up.

_______________________________________________________________________

Related Links:

BYJU'S

Desmos

CueMath

Downloadable Practice Problems.

Solutions to Practice Problems

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Area and Perimeter of Basic Geometric Shapes

Monday, April 13, 2020

Area and Perimeter of Basic Geometric Shapes

Geometric shapes can be found in basic math classes. Teachers usually presents circles, rectangles, squares, triangles, and the like. Then we are asked to find the Area and Perimeters of such. Then what we missed always for the final answers are the units. In this page, we are presenting again some same basics of it. This might be easy for those who advanced but might helpful for those who is not really engage in math subjects or beginners. This can be the good page for review our basic geometrical information. Well, area is simply getting the entire plane of the object, while perimeter the addition of all sides of the said object. Let's go.

Illustrative Examples:

1. ) Find the Area and Perimeter of the given figure below.

Given: width (w) = 3 in.
             length (l) = 5 in.

Required: Area and Perimeter

Solution: Area, taking all surface

Formula of Area of the rectangle: A = lw

A = lw

    = 5 in. x 3 in. = 15 in. , answer


Solution: Perimeter, sum of all sides


Formula for the Perimeter of the rectangle: P = 2(l + w)

P = 2(5 in. + 3 in.)

    = 2 ( 8 in.) = 16 in. , answer




2. ) Find the Area and Perimeter of the given figure below. So this is the same rectangle flipped side way. We can think which side will be our length or width, it doesn't matter.

We choose 7 cm. as our width, and 11 cm. for our length.

Given: width (w) = 7 cm.
             length (l) = 11 cm.

Required: Area and Perimeter

Solution: Area, taking all surface

Formula of Area of the rectangle: A = lw

A = lw

    = 7 cm. x 11 cm. = 77 cm. , answer


Solution: Perimeter, sum of all sides

Formula for the Perimeter of the rectangle: P = 2(l + w)

P = 2(11 cm. + 7 cm. )


    = 2 ( 18 cm.) = 36 in. , answer

3. ) Find the Area and Perimeter of the given triangle below. So this is a triangle with given 
dimensions. If, just in case, the dimension is not given, we have to draw.

Figure 1.

In triangle, finding the slant side measurement needs special formula called Pythagorean Theorem, and the formula is found to be;

Where the c = hypotenuse side, a = adjacent side, and b= opposite side. See figure below. (see the theta for angle reference).

Figure 2.

So how to getting the slant side of the triangle in figure 2 is by the use of the said theorem, so we have;

 Given a = 4

            b = 1.5, (half of 3)

            

To solve for the value of c;

            

            

            

             

              

               ft.

Now the dimension of the slant side.

To solve;

Given: Height (h) = 4 ft.
             Base (b) = 3 ft.
             Hypotenuse side = 4.27 ft.

Required: Area and Perimeter

Solution: Area, taking all surface

Formula for the Area of a triangle: A = 1/2(bh),   (b = base, h =height)

A = 1/2(bh)

    = 1/2(3 ft. x 4 ft.)

    =1/2(7 sq. ft.), (sq = square, ft. feet)

    = 3.5 square  feet. , answer


Solution: Perimeter, sum of all sides

Formula for the Perimeter of a triangle: P = s1 + s2 +s3,  (here, s1 = s2)

P = 4.27 ft. + 4.27 ft. + 3 ft.

    = 11.54 ft. , answer



4. ) Find the Area and Circumference of the given circle below. We knew that circle has no perimeter, so we deal with the circumference instead.

Illustration:

So,

Given: Diameter (d) = 10 in.
             Radius (b) = 5 in.,   (r = diameter /2)
      
Required: Area and Circumference

Solution: Area, taking all surface

Formula for the Area of a circle: ,   r = radius







 . answer


or, if diameter is used;

Formula for the Circumference of a circle: for diameter

,   d = diameter





   or

 . answer

Formula for the Circumference of a circle: for radius

 or

. answer

Example Problems:

1.) Given a rectangle with one side measures 10 in. and the other side 8 in., find the area and perimeter of the said object. (So we need to illustrate the statement).

Illustration:

Given: length (l) = 10 in.
             width (w) = 8 in.

Required: Area and Circumference

Solution: Area, taking all surface

Formula for the Area of a rectangle: Area = l*w

 Area = l*w

          = 10 in. x 8 in.

          = 80 sq. in. (sq. in = square inches), answer

Formula for the Perimeter of a rectangle: P = 2(l + w)

P = 2(l + w)

   = 2 (10 in. + 8 in.)

   = 2 (18 in.)

   = 36 in., answer


2.) A triangle has the given dimension below, see figure.

Figure:

Find the Area and Perimeter of the given triangle:

Given: Height (h) = 12 ft.
             Base (b) = 9ft.

Required: Area and Perimeter

Solution: Area, taking all surface

Formula for the Area of a triangle: Area = 1/2 b*h

Area = 1/2 b*h

         = 1/2 (9 ft.*12 ft.)

         = 1/2 (108 sq. ft.) 

         = 54 sq. ft. , answer


We need to find the slant side(hypotenuse first before getting the Perimeter

  where a = 9 ft., b =12 ft.









, answer (So the hypotenuse (slant side) is 15 ft.)

Formula for the Perimeter of a triangle:

P = S1 +S2 +S3

P = 9 ft. + 12 ft. + 15 ft.

P = 36 ft., answer


3.) A circle has radius measures 7 cm. Find its area and Circumference. (So we need to illustrate the statement).

Illustration:

Given: radius (r) = 7 cm.
            
Required: Area and Circumference

Solution: Area, taking all surface

Formula: 








Circumference Formula:







  answer

 4. Find the Area and Perimeter of the square whose side measures 2 meters.

So we have to illustrate the figure knowing it is not visually given.

Illustration:

Given: side (s) = 2 m.
            
Required: Area and Perimeter

Solution: Area, taking all surface

Formula for Area of a square: , where s = side

, answer

Formula for Perimeter of a square:  , where s = side





, answer


Let's take a pop Quiz:

1.) Find the Area and the perimeter of the given figure below.

Illustration:

Area is: click to write your answer

Perimeter is: click to write your answer


2.) What is the Area and Perimeter of the given triangle below?

Area is: click to write your answer

Perimeter is: click to write your answer

3.) Find the Area and Circumference of the circle whose radius measures 4 inches.

"here you have to illustrate the statement"

Area is: click to write your answer

Circumference is: click to write your answer

Quiz:

1.) The top of the table measures 4 ft. by 6 ft. .What is the area of the table and its Perimeter?

"here you have to illustrate the statement"

Area is: ___________or click

Perimeter is: ___________or click



2.) Given the figure below, find its Area and Perimeter.

figure:

Area is: ___________ or click

Perimeter is: ___________or click


3.) The standard size of a Basket ball ring is 18 inches in diameter. Find its Area and Circumference.

"here you have to illustrate the statement"

Area is: ___________ or click

Circumference is: ___________or click

4.) Find the total Area and Perimeter of the given figure below.

Figure:

Total Area is: _________ or click

Perimeter is: _________ or click


5.) Find the total Area of the given figure below.

Total Area is: _________ or click

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