Hungary Math Olympiad Problem | Best Math Olympiad Problems | Geometry Problem

Math Booster
29 Dec 202316:31
EducationalLearning
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TLDRThis video script explores a geometric problem involving a circle inscribed within a rectangle, with tangents of varying lengths from points A, B, C, and D. The challenge is to determine the value of 'X'. The solution uses the properties of tangents being perpendicular to the radius at the point of tangency and applies Pythagoras' theorem to set up equations. By comparing and subtracting these equations, the script derives the value of 'X'. The process involves algebraic manipulation and understanding of geometric relationships, concluding with the calculation of 'X' as the square root of 7361, emphasizing the importance of considering only the positive value due to the physical context of length.

Takeaways
  • πŸ“ The problem involves a circle inscribed in a rectangle with tangents from points A, B, C, and D to the circle.
  • β­• The length of the tangents from points A, B, C, and D to the circle are given as 33, 85, x, and 35 respectively.
  • βŠ₯ The script explains that the radius and tangent to a circle are perpendicular to each other.
  • πŸ”Ά The center of the circle is denoted as O, and angles formed by the radius and tangents are 90Β°.
  • πŸ“ The script uses Pythagoras' theorem to express the lengths of segments AO, BO, CO, and DO in terms of the radius (r) and the tangent lengths.
  • πŸ” By joining points A, B, C, and D to the center O, the script introduces points M and N to further analyze the geometry.
  • πŸ“ The script introduces variables H1 and H2 to represent the lengths of segments AS and SD, respectively.
  • πŸ“Š The script sets up equations for the squares of the lengths of segments AO, BO, CO, and DO based on the geometry of the rectangle and the circle.
  • πŸ”’ The script derives four equations by comparing the lengths of segments and uses them to find the value of x.
  • 🧩 By subtracting these equations, the script simplifies the problem to find the value of xΒ².
  • πŸ“ The final step involves calculating xΒ² by adding and subtracting the appropriate terms and then taking the square root to find the positive value of x.
Q & A
  • What is the main problem presented in the video?

    -The main problem is to find the length of the tangent (x) from point C to a circle that is inscribed inside a rectangle, given the lengths of other tangents from points A, B, and D.

  • What property of a circle is used in the video to solve the problem?

    -The property that a radius and a tangent to a circle are perpendicular to each other is used.

  • How many angles are mentioned to be 90Β° in the video?

    -Four angles are mentioned to be 90Β°, which are the angles formed between the tangents and the radii at the points of contact.

  • What is the significance of point M in the video?

    -Point M is used to create a right triangle by joining AO and OM, which helps in applying the Pythagorean theorem.

  • What is the relationship between AO and OM in the video?

    -AO is the length of the tangent from point A, and OM is the radius of the circle.

  • What are the equations derived for AO, BO, and CO in terms of s and r?

    -The equations are AO = √(AO^2) = √(33^2 + r^2), BO = √(BO^2) = √(85^2 + r^2), and CO = √(CO^2) = √(X^2 + r^2).

  • How does the video relate the rectangle to the circle and the tangents?

    -The video uses the rectangle's sides and the perpendicular lines from the center O to the rectangle's sides to form right triangles with the tangents.

  • What are the variables H1 and H2 used for in the video?

    -H1 and H2 are the lengths of the perpendicular segments from the center O to the sides of the rectangle, which are used in the equations for the tangents.

  • How many equations are formed in the video to solve for x?

    -Four equations are formed, one for each tangent from points A, B, C, and D.

  • What mathematical operation is used to isolate x in the equations?

    -Subtraction of equations is used to isolate x and simplify the equations to find its value.

  • How does the video conclude the value of x?

    -By subtracting and simplifying the equations, the video finds that x^2 equals 7361, and thus x is the square root of 7361, which is approximately 86.

Outlines
00:00
πŸ“ Geometric Relationships in a Circle and Rectangle

This paragraph introduces a geometric problem involving a circle inscribed within a rectangle with tangents from points A, B, C, and D to the circle, each with different lengths: 33, 85, x, and 35 respectively. The script explains that tangents to a circle are perpendicular to the radius at the point of tangency, forming right angles. By drawing lines from the center of the circle (O) to the points of tangency and applying the Pythagorean theorem, the script sets up equations to solve for x, the unknown length of the tangent from point C.

05:01
πŸ“ Constructing Perpendiculars and Forming Equations

The second paragraph continues the problem-solving process by drawing perpendicular lines from the center of the circle (O) to the sides of the rectangle, creating additional points P, Q, S, and T. It introduces variables B1, B2, H1, and H2 to represent segments of these perpendiculars and uses these to form equations based on the Pythagorean theorem. The script then equates these equations to the lengths of the tangents, creating a system of equations to solve for x.

10:02
πŸ” Solving for x by Subtracting Equations

In this paragraph, the script simplifies the problem by subtracting the equations derived from the geometric setup. This process eliminates variables representing the radius of the circle (r) and the segments H1 and H2, leaving equations that relate the squares of the lengths of the sides of the rectangle (B1, B2) to the squares of the tangent lengths (33, 85, x, 35). The script identifies two key equations (5 and 6) that will be used to solve for x.

15:08
πŸ”’ Calculating the Value of x

The final paragraph concludes the problem by solving for x. It uses the equations derived in the previous paragraph to find that xΒ² equals the difference between the squares of the lengths of the tangents (85 - 33 + 35). The script then performs arithmetic operations to calculate the value of xΒ² as 7225. It explains the process of finding the square root to determine the positive value of x, which is a length and must be positive. The final answer for x is given as the square root of 7225.

Mindmap
Keywords
πŸ’‘Circle
A circle is a geometric shape consisting of all points equidistant from a given point called the center. In the video's theme, the circle is nested inside a rectangle, and the problem revolves around finding the length of a tangent from various points on the circumference of the circle. The circle's properties, such as its radius and the perpendicularity of tangents to it, are central to solving the geometric problem presented.
πŸ’‘Rectangle
A rectangle is a quadrilateral with four right angles and opposite sides equal in length. In the context of the video, a rectangle contains a circle, and the problem involves geometric relationships between the circle and the rectangle's sides. The rectangle provides the outer boundary within which the circle and its tangents are considered.
πŸ’‘Tangent
A tangent to a circle is a line that touches the circle at exactly one point and is perpendicular to the radius at the point of contact. In the video script, tangents of lengths 33, 85, x, and 35 are mentioned, and the task is to find the value of x, the length of an unknown tangent. The concept of tangency is crucial for applying geometric theorems to find the missing length.
πŸ’‘Perpendicular
Perpendicularity refers to the property of two lines or planes that intersect at an angle of 90 degrees. In the video, it is mentioned that the tangents to the circle are perpendicular to the radii at the points of tangency. This property is fundamental in establishing right angles in the geometric figures considered, which is essential for applying the Pythagorean theorem.
πŸ’‘Radius
The radius of a circle is the distance from the center of the circle to any point on its circumference. In the script, the radius (denoted as 'r') is used in the Pythagorean theorem to relate the lengths of the tangents and the segments of the radius. The radius is a key element in the geometric relationships and equations derived from the circle and its tangents.
πŸ’‘Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In the video, this theorem is used to establish relationships between the lengths of the tangents, the radii, and the segments created by the tangents intersecting the circle's radius. It is a cornerstone in solving for the unknown length x.
πŸ’‘Angle
An angle is the figure formed by two rays sharing a common endpoint, known as the vertex. In the video, angles are mentioned in the context of right angles formed by the perpendicularity of tangents to the circle and the radii. The angles are integral to understanding the geometric configuration and applying the Pythagorean theorem.
πŸ’‘Equation
An equation is a mathematical statement that asserts the equality of two expressions. In the script, equations are formed based on the geometric relationships and the Pythagorean theorem to represent the lengths of the tangents and segments of the radius. These equations are crucial for setting up the problem and finding the value of x.
πŸ’‘Length
Length is a measure of the extent of something from end to end. In the video, the problem involves finding the length of a tangent (denoted as 'x'). Lengths of other tangents are given as 33, 85, and 35, and the script uses these lengths to derive equations and solve for the unknown length x.
πŸ’‘Square Root
A square root of a number is a value that, when multiplied by itself, gives the original number. In the video, the square root is used in the final step to solve for x, as the equations lead to finding the square of x. The square root is essential in extracting the positive value of x, which is a physical length and must be positive.
Highlights

A circle is inscribed inside a rectangle with tangent lengths from points A, B, C, and D being 33, 85, x, and 35 respectively.

The radius and tangent of a circle are perpendicular to each other.

Angles formed by the tangents and the radius are all 90Β°.

By joining the center of the circle to the points of tangency, we can use Pythagoras' theorem to find the lengths.

The length AO is expressed as the square root of the sum of the squares of AM and OM, where OM is the radius.

Similar expressions are derived for lengths BO, CO, and DO using the same method.

A rectangle ABCD is considered with a circle inscribed within it, and perpendicular lines are drawn from the center through points P, Q, S, and T.

The lengths of AP, BP, DQ, and QC are denoted as B1, and lengths AS, SD as H1 and H2.

Expressions for lengths AO, BO, CS, and DS are derived using the right-angled triangles formed.

Four equations are established based on the derived expressions for AO, BO, CS, and DS.

By comparing and subtracting these equations, relationships between B1, B2, H1, H2, and the unknown x are found.

Equations 5 and 6 are derived to relate B2 squared minus B1 squared to the known tangent lengths and x squared.

The value of x squared is calculated by adding and subtracting the squares of the known tangent lengths.

The final value of x squared is found to be 7361 after simplification.

The length x is determined to be the positive square root of 7361, as length cannot be negative.

The process involves algebraic manipulation and application of geometric principles.

The solution requires understanding of right-angled triangles and the Pythagorean theorem.

The video demonstrates a methodical approach to solving geometry problems involving circles and tangents.

Transcripts
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