Khan Academy Exercises and Videos

Inscribe and circumscribe a triangle

2026-05-07T12:00:39.408441

Learn how to inscribe and circumscribe a triangle similar to a given triangle in and about a circle with clear, step‑by‑step construction. In this lesson, we not only show the geometric steps but also justify each move with appropriate proofs, ensuring you understand both the process and the reasoning behind it. With diagrams and logical explanations, you’ll see how similarity, circle properties, and triangle geometry come together to make the construction precise and meaningful.

Common Tangent

2026-05-07T12:00:39.408441

Explore the geometry of tangent circles in this detailed lesson! We cover the different types of common tangents for non‑intersecting circles, tangent circles, and circles intersecting at two points. You’ll also learn an important theorem about the relationship between the centers of tangent circles and their point of contact, along with its corollaries. With clear explanations and diagrams, this video makes the concept of tangent circles and their tangents easy to understand and apply in problem‑solving.

Congruence arcs and chords

2026-05-07T12:00:39.408441

If two arcs in a circle are the same size, must their chords be equal too? And if two chords are equal, does that force the arcs to match? This video proves both directions of this beautiful relationship step by step. You will learn how arcs, chords, and central angles are all linked within a circle, and how the SAS and SSS congruence rules from triangles unlock two powerful theorems about circles. Follow along as we use a radius of 5 cm and central angles of 60 degrees to show why congruent arcs must create equal chords, then flip the argument to show that equal chords of 8 cm force their arcs to be congruent as well. The video also covers an important constraint: these theorems only hold within the same circle or in congruent circles. A challenge question at the end ties both theorems together.

Cross Multiplication Method

2026-05-07T12:00:39.408441

Can you solve a pair of simultaneous equations in under 30 seconds? With the cross multiplication method, a single formula gives you the values of x and y directly without working through substitution or elimination step by step. This video teaches you the complete cross multiplication formula x / (b1c2 - b2c1) = y / (c1a2 - c2a1) = 1 / (a1b2 - a2b1) and shows you exactly how to label coefficients correctly, especially getting the signs right when moving constants to the left side. You will also learn a simple b-c-a cycle trick to memorise the formula in seconds, follow a full worked example using 3x + 2y = 12 and x - 4y = 2, and understand what happens when the denominator turns out to be zero, covering both parallel and coincident lines.

Cramer's Rule (2x2)

2026-05-07T12:00:39.408441

Want to solve simultaneous equations in seconds, entirely in your head? This video introduces Cramer's Rule, a powerful shortcut that uses determinants instead of elimination or substitution to find x and y. If you have ever wished for a faster way to solve a pair of linear equations, this method will change everything. We start with the basics of what a determinant is and how to calculate it from a 2x2 matrix by multiplying diagonals and subtracting. Then we walk through the complete process of building three key determinants: Delta, Delta x, and Delta y. You will learn exactly which column to replace with the constants and why that pattern matters. Follow along with two fully worked examples, including step by step verification of the answers (3, -2) and (1, 2). We also cover the most common mistake students make and what happens when Delta equals zero, giving infinitely many solutions instead of a unique one.

Trigonometric Ratios for Obtuse angles

2026-05-07T12:00:39.408441

In this lesson, we explore how trigonometric ratios extend beyond right‑angled triangles. Since right‑angled triangles cannot represent obtuse angles, we shift to the coordinate plane to define and calculate trigonometric ratios for angles greater than 90°. Step by step, we uncover the rules for the second quadrant, the special case at 180°, and the relationships between acute angles and their supplementary or related obtuse angles. Finally, we apply these results to find exact values at 120°, 135°, and 150°.

Construct: Incircle, circumcircle to square

2026-05-07T12:00:39.408441

Learn how to draw the circumcircle and incircle of a square step by step! In this lesson, we not only carry out the construction but also explain the logic behind each move — why we choose certain points, how symmetry guides the process, and what geometric properties ensure accuracy. With simple reasoning and clear diagrams, you’ll understand both the “how” and the “why” of these constructions, making it easier to apply them in exams and classroom practice.

Two Circles in a Plane

2026-05-07T12:00:39.408441

Discover the different ways two circles can lie in a plane in this lesson! We explore non‑intersecting circles, circles that touch at a single point, and circles that intersect each other at two points. With simple explanations and diagrams, this video makes the geometry of circles easy to visualize and understand, helping you grasp the key differences between these cases.

Construct circumscribing circle

2026-05-07T12:00:39.408441

Learn how to circumscribe an equilateral triangle, a square, and a regular hexagon about a given circle with step‑by‑step construction. In this lesson, we explain not only the drawing process but also the geometric logic behind each step — why certain points are chosen, how symmetry ensures accuracy, and what properties make the construction valid. With clear reasoning and diagrams, this video makes the topic easy to follow and perfect for exam preparation or classroom practice.

Construct inscribing circle

2026-05-07T12:00:39.408441

Learn how to inscribe regular polygons in a circle with step‑by‑step construction! In this lesson, we begin with the basics of inscribing polygons and then show how to inscribe an equilateral triangle, a square, and a regular hexagon inside a given circle. We also explore an alternate method for inscribing an equilateral triangle and a regular hexagon, making the process easier to understand. With logical explanations and clear diagrams, this video helps you grasp both the construction steps and the reasoning behind them — perfect for exam preparation and classroom practice.

Compound angle formulas for tan, cot

2026-05-07T12:00:39.408441

In this video, we derive the compound angle formulas step by step, covering the following key identities: tan⁡(a±b) cot⁡(a±b)

Relation between roots and coefficients

2026-05-07T12:00:39.408441

Did you know you can find the sum and product of the roots of any quadratic equation without actually solving it? This video reveals the elegant relationship between the roots and the coefficients of ax squared + bx + c = 0. Starting from a concrete example with x squared - 7x + 12 = 0, you will see why the sum of roots equals -b/a and the product equals c/a, then follow a complete proof using the quadratic formula and the discriminant D. The video goes beyond the basics and shows you how to find alpha squared + beta squared, alpha cubed + beta cubed, and alpha/beta + beta/alpha using algebraic identities alone, all without knowing the individual roots. You will also learn how to reconstruct a quadratic equation if only the sum and product of its roots are given, and see the most common sign error students make and how to avoid it.

Draw circumcircle of a triangle

2026-05-07T12:00:39.408441

In this video, we learn how to draw the circumcircle of a triangle when the length of one side and the measure of the angle opposite to it are given. We begin by discussing the logic behind the construction, ensuring you understand why each step works. Then, we carefully go through the construction process for three different cases—when the angle is acute, when it is a right angle (90°), and when it is obtuse. This lesson blends clear reasoning with practical steps, making the concept of circumcircles easy to grasp.

Arithmetic Means

2026-05-07T12:00:39.408441

In this video, we dive into the concept of the Arithmetic Mean (A.M.). We begin by understanding what the arithmetic mean is and how to calculate it between two numbers. Then, we explore its geometric meaning to build intuition. Step by step, we learn how to insert 2 A.M.s, 3 A.M.s, and finally n A.M.s between two given numbers. This lesson combines clear explanations with worked examples so you can master the topic and apply it confidently in problem‑solving.

Cancellation Laws

2026-05-07T09:40:36.411628

In this video, we explore the cancellation laws of addition and multiplication in the set of real numbers. Step by step, we prove how these laws follow directly from the axioms of real numbers. We then extend the discussion to additional corollaries involving zero, negatives, and inverses, showing how they strengthen our understanding of algebraic operations. Finally, we apply these results to rigorously prove an algebraic identity using the axioms of real numbers, demonstrating how foundational principles lead to powerful results in algebra.

Symmetric Difference

2026-05-07T09:40:36.411628

In this video, we dive into the concept of the symmetric difference of sets and explore it step by step. We begin by explaining what symmetric difference means and how it can be visualized using a Venn diagram. Then, we work through examples to make the idea concrete and easy to understand. Finally, we discuss important properties of symmetric difference, including commutativity and associativity, showing how these rules make set operations more powerful and systematic.

Continued Fraction

2026-05-07T09:40:36.411628

Learn continued fractions step by step! We start from the bottommost fraction and move upward, showing exactly how the expansion works. With clear solved examples, you’ll see how continued fractions are built and applied in problem‑solving, making the concept easy to understand and practice.

HCF of Polynomials

2026-05-07T09:40:36.411628

Master the skill of finding the HCF of polynomials with ease! In this lesson, we explain the step‑by‑step process of factoring polynomials and identifying their highest common factor. With clear solved examples drawn from exam‑style problems, you’ll learn how to break down expressions, spot common terms, and simplify systematically.

Laws: Trichotomy & Transitivity

2026-05-07T09:40:36.411628

In this lesson, we explore the laws of order in the set of real numbers, including the law of trichotomy and the law of transitivity. We then extend these foundational axioms to prove important results step by step. Using clear reasoning, we demonstrate how these laws help establish inequalities and strengthen our understanding of algebra. Finally, we apply them to prove two key theorems involving inequalities with real numbers.

Area of general quadrilateral

2026-05-07T07:40:40.092770

In this video, we explore different methods to calculate the area of a general quadrilateral using three distinct approaches. First, we learn how to find the area when one diagonal and the perpendiculars drawn from the opposite vertices are given. Next, we examine the special case where the diagonals bisect each other at right angles. Finally, we apply Heron’s formula to determine the area when all four sides and one diagonal are known. Each method is explained step by step to help you understand the concepts thoroughly and apply them with confidence in problem-solving.

Absolute value of a rational number

2026-05-08T07:51:30.000000

Estimate multi-digit division problems

2026-05-07T16:39:53.000000

Use estimation to find a reasonable solution to multi-digit division problems.

The Cold War

2026-05-07T14:33:45.000000

Practice what you learned in lesson 8.8: The Cold War!

Trigonometric identities challenge problems

2026-05-07T11:50:02.000000

Let's practice all the identities we've discussed so far.

Square roots using prime factorisation (advanced)

2026-05-07T07:05:06.000000

Square roots using prime factorisation (advanced)

Computation of Income Tax

2026-05-07T06:18:36.000000

In this exercise, students will be able to calculate annual income tax for different age categories by applying progressive tax slabs and permissible deductions to arrive at the final tax liability.

Apply: Alpha, beta, and gamma decay

2026-05-06T19:46:11.000000

Apply your understanding of alpha, beta, and gamma decay in this set of free practice questions.

Apply: electric motors

2026-05-06T19:43:59.000000

Apply your knowledge of electric motors in this set of free practice questions.

Angular momentum

2026-05-06T19:34:59.000000

Check your understanding of angular momentum in this set of free practice questions.

Prove triangle properties

2026-05-06T16:52:51.000000

Use transformations, line and angle relationships, and triangle congruence criteria to prove properties of triangles.

Find angles in congruent triangles

2026-05-06T16:47:12.000000

Given two triangles, determine whether they are congruent and use that to find missing angle measures.

The Holocaust

2026-05-06T15:12:33.000000

Practice what you learned in lesson 8.7: The Holocaust.

HCF of Polynomials

2026-05-06T07:34:57.000000

In this exercise, students will be able to apply factorization techniques, including common factor extraction and algebraic identities, to determine the Highest Common Factor (HCF) of single-variable, multi-variable, and sets of three polynomials.

Income Tax

2026-05-06T07:29:00.000000

In this exercise, students will be able to define core taxation terminology and identify the specific legal frameworks, age-based classifications, and deduction limits that govern individual income tax liability.

Create histograms

2026-05-06T05:06:11.000000

Practice creating histograms.

Telling time word problems (within the hour)

2026-05-06T05:05:17.000000

Solve a word problem to find the duration of an event. Both analog or digital clocks are included.

Identify quadrilaterals

2026-05-06T05:04:37.000000

Identify squares, rectangles, and rhombuses.

Understand: Parts of eukaryotic cells

2026-05-05T18:39:53.000000

Check your understanding of eukaryotic cell parts in this set of free, standards-aligned practice questions.

Vector word problems

2026-05-05T18:04:12.000000

Apply what you've learned about vectors to solve some word problems!

Polar & rectangular forms of complex numbers

2026-05-05T17:57:04.000000

Given a complex number in rectangular form, write it in polar form. Given a complex number in polar form, write it in rectangular form.