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Learn GCSE Maths: Easy Median and Circle Theorems

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Learn GCSE Maths: Easy Median and Circle Theorems
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Tanushree

@tanushree_bait

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Verified Study Note

This document covers various mathematical concepts, focusing on GCSE maths topics including median and mean from grouped frequency tables, angle theorems, quadratic equations, trigonometry, and circle theorems. It provides detailed explanations, formulas, and visual representations to aid understanding.

Key points:

  • Calculation methods for median and mean from grouped frequency tables
  • Angle relationships in geometry, including corresponding, co-interior, and alternate angles
  • Quadratic equation solving techniques
  • Trigonometric ratios and graphs
  • Circle theorems and their applications

17/06/2023

495

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Angle Relationships in Geometry

This section covers fundamental angle relationships in geometry, which are essential for solving various GCSE maths problems.

The page illustrates three key angle relationships:

  1. Corresponding angles
  2. Co-interior angles
  3. Alternate angles

Definition: Corresponding angles are equal angles that appear in the same relative position when a line intersects two other lines.

Highlight: Understanding these angle relationships is crucial for solving complex geometric problems and proofs.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Vertically Opposite Angles

This page focuses on vertically opposite angles, another important concept in geometry and GCSE maths.

The diagram illustrates that vertically opposite angles are equal. This principle is demonstrated with angles labeled as x and 180-x.

Definition: Vertically opposite angles are pairs of angles opposite each other when two lines intersect.

Highlight: The equality of vertically opposite angles is a fundamental principle used in many geometric proofs and problem-solving scenarios.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Quadratic Equations: Completing the Square

This page introduces the method of completing the square for solving quadratic equations, an important technique in GCSE maths.

The process of completing the square is demonstrated step-by-step, showing how to transform a quadratic equation into the form (x + p)² + q.

Example: For the equation x² + 4x + 7 = 0, the completed square form is (x + 2)² - 4 + 7 = 0.

Highlight: Completing the square is useful for finding the minimum point of a quadratic function, which in this case is (-2, 3).

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Quadratic Equations: Factorization

This page covers the factorization method for solving quadratic equations, another essential skill in GCSE maths.

The process of factoring a quadratic equation into two brackets is demonstrated, using the example x² - 2x - 15.

Example: x² - 2x - 15 factorizes to (x - 5)(x + 3), giving roots at x = 5 and x = -3.

Highlight: Factorization allows for easy identification of the roots (x-intercepts) of a quadratic function, which are (5, 0) and (-3, 0) in this case.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Transformations of Functions

This page explains various transformations of functions, an important topic in GCSE maths that relates to graph sketching and manipulation.

Different transformations are illustrated:

  • y = f(x - 2): Shift 2 units right
  • y = 3f(x): Stretch vertically by a factor of 3
  • y = f(x) - 2: Shift 2 units down
  • y = f(-x): Reflection in the y-axis
  • y = f(2x): Horizontal compression by a factor of 1/2
  • y = -f(x): Reflection in the x-axis

Highlight: Understanding these transformations is crucial for sketching and interpreting graphs of various functions.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Trigonometric Ratios

This page presents a diagram showing the trigonometric ratios for common angles (30°, 45°, 60°, 90°) in a right-angled triangle.

The ratios of sine, cosine, and tangent are provided for these angles.

Vocabulary: Sine (sin), cosine (cos), and tangent (tan) are the main trigonometric ratios used in GCSE maths.

Highlight: Memorizing these common angle ratios is essential for solving trigonometric problems efficiently.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Circle Equation

This page introduces the general equation of a circle, an important concept in coordinate geometry and GCSE maths.

The equation (x - a)² + (y - b)² = r² is presented, where:

  • (a, b) represents the center of the circle
  • r² is the square of the radius

Definition: This equation describes all points (x, y) that lie on a circle with center (a, b) and radius r.

Highlight: Understanding this equation is crucial for solving problems involving circles in coordinate geometry.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Circle Theorems

This page provides a comprehensive overview of circle theorems, which are fundamental to GCSE maths and geometry.

The page covers several important theorems:

  1. Alternate segment theorem
  2. Angles in the same segment theorem
  3. Perpendicular from center to chord theorem
  4. Angle at the center theorem
  5. Angles in a semicircle theorem
  6. Cyclic quadrilateral theorem
  7. Tangent-radius theorem

Example: The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.

Highlight: These theorems are essential for solving complex geometric problems involving circles and are frequently tested in GCSE maths exams.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Sine Function Graph

This page illustrates the graph of y = sin(x), a fundamental trigonometric function in GCSE maths.

Key features of the sine graph are highlighted:

  • The graph starts at (0, 0)
  • It cycles through the values 1, 0, -1, 0 every 90°
  • The period of the function is 360°

Definition: The sine function relates the angle of a right-angled triangle to the ratio of the opposite side to the hypotenuse.

Highlight: Understanding the shape and properties of the sine graph is crucial for solving trigonometric equations and modeling periodic phenomena.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

View

Cosine Function Graph

This page shows the graph of y = cos(x), another important trigonometric function in GCSE maths.

Key features of the cosine graph are noted:

  • The graph starts at (0, 1)
  • It cycles through the values 0, -1, 0, 1 every 90°
  • The period of the function is 360°

Definition: The cosine function relates the angle of a right-angled triangle to the ratio of the adjacent side to the hypotenuse.

Highlight: The cosine graph is similar to the sine graph but shifted by 90°, which is important to recognize in trigonometric problems.

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Lena, iOS user

I love this app ❤️ I actually use it every time I study.

Subjects

/

Maths

/

Geometry and measure

/

Angles, lines and polygons

Learn GCSE Maths: Easy Median and Circle Theorems

user profile picture

Tanushree

@tanushree_bait

·

7 Followers

Follow

Verified Study Note

This document covers various mathematical concepts, focusing on GCSE maths topics including median and mean from grouped frequency tables, angle theorems, quadratic equations, trigonometry, and circle theorems. It provides detailed explanations, formulas, and visual representations to aid understanding.

Key points:

  • Calculation methods for median and mean from grouped frequency tables
  • Angle relationships in geometry, including corresponding, co-interior, and alternate angles
  • Quadratic equation solving techniques
  • Trigonometric ratios and graphs
  • Circle theorems and their applications

17/06/2023

495

 

11/10

 

Maths

10

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Angle Relationships in Geometry

This section covers fundamental angle relationships in geometry, which are essential for solving various GCSE maths problems.

The page illustrates three key angle relationships:

  1. Corresponding angles
  2. Co-interior angles
  3. Alternate angles

Definition: Corresponding angles are equal angles that appear in the same relative position when a line intersects two other lines.

Highlight: Understanding these angle relationships is crucial for solving complex geometric problems and proofs.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Vertically Opposite Angles

This page focuses on vertically opposite angles, another important concept in geometry and GCSE maths.

The diagram illustrates that vertically opposite angles are equal. This principle is demonstrated with angles labeled as x and 180-x.

Definition: Vertically opposite angles are pairs of angles opposite each other when two lines intersect.

Highlight: The equality of vertically opposite angles is a fundamental principle used in many geometric proofs and problem-solving scenarios.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Quadratic Equations: Completing the Square

This page introduces the method of completing the square for solving quadratic equations, an important technique in GCSE maths.

The process of completing the square is demonstrated step-by-step, showing how to transform a quadratic equation into the form (x + p)² + q.

Example: For the equation x² + 4x + 7 = 0, the completed square form is (x + 2)² - 4 + 7 = 0.

Highlight: Completing the square is useful for finding the minimum point of a quadratic function, which in this case is (-2, 3).

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Quadratic Equations: Factorization

This page covers the factorization method for solving quadratic equations, another essential skill in GCSE maths.

The process of factoring a quadratic equation into two brackets is demonstrated, using the example x² - 2x - 15.

Example: x² - 2x - 15 factorizes to (x - 5)(x + 3), giving roots at x = 5 and x = -3.

Highlight: Factorization allows for easy identification of the roots (x-intercepts) of a quadratic function, which are (5, 0) and (-3, 0) in this case.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Transformations of Functions

This page explains various transformations of functions, an important topic in GCSE maths that relates to graph sketching and manipulation.

Different transformations are illustrated:

  • y = f(x - 2): Shift 2 units right
  • y = 3f(x): Stretch vertically by a factor of 3
  • y = f(x) - 2: Shift 2 units down
  • y = f(-x): Reflection in the y-axis
  • y = f(2x): Horizontal compression by a factor of 1/2
  • y = -f(x): Reflection in the x-axis

Highlight: Understanding these transformations is crucial for sketching and interpreting graphs of various functions.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Trigonometric Ratios

This page presents a diagram showing the trigonometric ratios for common angles (30°, 45°, 60°, 90°) in a right-angled triangle.

The ratios of sine, cosine, and tangent are provided for these angles.

Vocabulary: Sine (sin), cosine (cos), and tangent (tan) are the main trigonometric ratios used in GCSE maths.

Highlight: Memorizing these common angle ratios is essential for solving trigonometric problems efficiently.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Circle Equation

This page introduces the general equation of a circle, an important concept in coordinate geometry and GCSE maths.

The equation (x - a)² + (y - b)² = r² is presented, where:

  • (a, b) represents the center of the circle
  • r² is the square of the radius

Definition: This equation describes all points (x, y) that lie on a circle with center (a, b) and radius r.

Highlight: Understanding this equation is crucial for solving problems involving circles in coordinate geometry.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Circle Theorems

This page provides a comprehensive overview of circle theorems, which are fundamental to GCSE maths and geometry.

The page covers several important theorems:

  1. Alternate segment theorem
  2. Angles in the same segment theorem
  3. Perpendicular from center to chord theorem
  4. Angle at the center theorem
  5. Angles in a semicircle theorem
  6. Cyclic quadrilateral theorem
  7. Tangent-radius theorem

Example: The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.

Highlight: These theorems are essential for solving complex geometric problems involving circles and are frequently tested in GCSE maths exams.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Sine Function Graph

This page illustrates the graph of y = sin(x), a fundamental trigonometric function in GCSE maths.

Key features of the sine graph are highlighted:

  • The graph starts at (0, 0)
  • It cycles through the values 1, 0, -1, 0 every 90°
  • The period of the function is 360°

Definition: The sine function relates the angle of a right-angled triangle to the ratio of the opposite side to the hypotenuse.

Highlight: Understanding the shape and properties of the sine graph is crucial for solving trigonometric equations and modeling periodic phenomena.

Median from arouped frequency table intervals freq cumu freq 3 3 6 9 6 0322210 104x420 20<x<30 304x440 40<x<50 Total: total frequency 2 = 12

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Cosine Function Graph

This page shows the graph of y = cos(x), another important trigonometric function in GCSE maths.

Key features of the cosine graph are noted:

  • The graph starts at (0, 1)
  • It cycles through the values 0, -1, 0, 1 every 90°
  • The period of the function is 360°

Definition: The cosine function relates the angle of a right-angled triangle to the ratio of the adjacent side to the hypotenuse.

Highlight: The cosine graph is similar to the sine graph but shifted by 90°, which is important to recognize in trigonometric problems.

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Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the #1 education app in five European countries

4.9+

Average app rating

15 M

Pupils love Knowunity

#1

In education app charts in 12 countries

950 K+

Students have uploaded notes

Still not convinced? See what other students are saying...

iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

Philip, iOS User

The app is very simple and well designed. So far I have always found everything I was looking for :D

Lena, iOS user

I love this app ❤️ I actually use it every time I study.