Can you find total area of the Pink circles? | (Rectangle) | #math #maths | #geometry
TLDRIn this geometry lesson, the video explores the problem of finding the total area of two pink circles inscribed within a rectangle, alongside two larger blue circles. Given the area of the blue circles and the rectangle's diagonal, the video uses the Pythagorean theorem and circle theorem to determine the radius of the pink circles, calculating their total area to be 81π/50 cm², approximately 5.09 cm².
Takeaways
- 📏 The video discusses a geometry problem involving two blue and two pink circles inscribed in a rectangle.
- 🔵 The area of each blue circle is given as 5 cm², and they are tangent to each other.
- 📐 The diagonal of the rectangle (AB) is 10 cm, which helps in determining the dimensions of the rectangle.
- 🔶 The task is to calculate the total area of the two pink circles.
- 📘 The radius of the blue circle (R) is determined to be 2 cm using the area formula (Area = πR²).
- 📍 By using the fact that the centers and points of tangency are colinear, the lengths of sides of the rectangle are deduced.
- 🔹 The right triangle ABD is used to find the lengths of sides AD and BD using the Pythagorean triplets (3, 4, 5).
- 📐 The horizontal and vertical lines drawn from the centers of the circles help in understanding the angles and lengths involved.
- 🔺 The radius of the pink circle (r) is found to be 9/10 cm using the Pythagorean theorem on triangle PQO.
- 📝 The area of a single pink circle is calculated using the formula (Area = πr²) and found to be 81π/100 cm².
- 🔴 The total area of both pink circles is the sum of the areas of the two individual circles, which is 81π/50 cm² or approximately 5.09 cm².
Q & A
What is the area of one blue circle mentioned in the video?
-The area of one blue circle is given as 4π cm².
How is the radius of the blue circle calculated in the script?
-The radius of the blue circle is calculated using the area formula for a circle (Area = πr²). By setting the area to 4π and solving for r, the radius is found to be 2 cm.
What geometric property is used to determine that the centers and points of tangency are colinear?
-The property used is that the centers and points of tangency of circles that are tangent to each other and a straight line lie on the same straight line.
What is the length of diagonal AB in the rectangle?
-The length of diagonal AB is given as 10 cm.
How are the lengths of sides AD and BC determined in the rectangle?
-By using the fact that the rectangle's sides are equal and the diagonal AB is 10 cm, and applying the Pythagorean triplets (3, 4, 5) multiplied by 2, the lengths of AD and BC are determined to be 8 cm each.
What is the radius of the pink circle calculated to be?
-The radius of the pink circle is calculated to be 9/10 cm, using the Pythagorean theorem on the right triangle formed by the radii and the distance between the centers of the blue and pink circles.
What is the total area of the two pink circles combined?
-The total area of the two pink circles is calculated to be 81π/50 cm², which is approximately 5.09 cm².
Why is the Pythagorean theorem applied to the triangle formed by the centers of the blue and pink circles and the point of tangency?
-The Pythagorean theorem is applied because the triangle is a right triangle, with the sides being the radii of the circles and the distance between the centers, which allows for the calculation of the unknown radius of the pink circle.
What is the significance of the 3-4-5 Pythagorean triplets in this problem?
-The 3-4-5 Pythagorean triplets are used to determine the lengths of the sides of the right triangle formed by the rectangle's sides and the diagonal, by scaling the triplets to match the given diagonal length.
How does the video ensure that the calculations are related to the actual figure?
-The video provides a disclaimer that the figure may not be 100% true to scale, emphasizing the importance of following the mathematical process rather than relying solely on the visual representation.
Outlines
📐 Introduction to the Geometry Problem
The video begins by introducing a geometry problem involving two sets of circles inscribed within a rectangle. The blue circles have an area of 5 cm² each, and the challenge is to find the total area of the two pink circles. The presenter emphasizes the tangency of the circles and the rectangle's diagonal length of 10 cm. The first step is to determine the radii of both the blue and pink circles, denoted as uppercase R and lowercase r, respectively. The area formula for a circle is used to calculate the radius of the blue circle, which is found to be 2 cm.
🔍 Analyzing the Rectangle and Applying Pythagorean Triplets
The presenter then focuses on the rectangle's dimensions, using the fact that the centers and points of tangency are colinear to deduce the lengths of the rectangle's sides. By recognizing the rectangle's sides as a scaled-up Pythagorean triple (3, 4, 5), the sides are calculated to be 6 cm and 8 cm. This step is crucial for understanding the spatial arrangement of the circles and for subsequent calculations.
📐 Solving for the Radius of the Pink Circles
In this paragraph, the presenter creates additional lines to emphasize the right angles formed by the radii and tangent lines. By connecting the centers of the blue and pink circles, a right triangle is formed, which is then used to apply the Pythagorean theorem. The presenter uses algebraic identities to simplify the equation and solve for the radius of the pink circle, which is found to be 9/10 cm. This step is key to finding the area of the pink circles.
🔢 Calculating the Total Area of the Pink Circles
The final paragraph concludes the problem by calculating the area of a single pink circle using the formula for the area of a circle with the radius found in the previous step. The area is simplified to 81π/100 cm². The total area for both pink circles is then found by doubling this value, resulting in 81π/50 cm², which approximates to 5.09 cm². The presenter ends the video with a call to action for viewers to subscribe for more content.
Mindmap
Keywords
💡Inscribed Circles
💡Tangent Circles
💡Area Calculation
💡Radius
💡Diagonal
💡Pythagorean Theorem
💡Colinear
💡Pythagorean Triplets
💡Circle Theorem
💡Simplification
Highlights
The video explores the total area of two pink circles inscribed within a rectangle.
Two identical blue circles and two pink circles are tangent to each other within the rectangle.
The area of each blue circle is given as 5 cm².
The diagonal of the rectangle (AB) is 10 cm.
The radius of the blue circle is denoted as uppercase R, and the pink circle as lowercase r.
The area formula for a circle (Area = π * r²) is used to find the radius of the blue circle.
The radius of the blue circle (R) is calculated to be 2 cm.
The rectangle's sides are determined to be 8 cm based on the blue circle's diameter.
A right triangle is identified within the rectangle using the Pythagorean triplets.
The side lengths of the right triangle are found to be 6 cm and 8 cm.
The theorem that the angle between a radius and a tangent line is 90° is applied.
A right triangle PQO is formed by connecting the centers of the pink and blue circles.
The radius of the pink circle (r) is found to be 9/10 cm using the Pythagorean theorem.
The area of a single pink circle is calculated to be 81π/100 cm².
The total area of both pink circles is found to be 81π/50 cm², approximately 5.09 cm².
The video concludes with a reminder to subscribe for more educational content.
