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Law of Cosines
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Today we're going to explore the Law of Cosines. It's handy when we want to find an unknown side in a triangle and we know the other two sides and the angle between them. Can anyone tell me how the Law of Cosines is expressed?
Is it like aยฒ = bยฒ + cยฒ - 2bc cos(A)?
Exactly! This formula connects the lengths of the sides with the cosine of the included angle. Letโs consider an example: suppose we have a triangle with sides a = 7, b = 10, and the angle A = 60ยฐ. What will be the length of side c?
We can plug those values into the formula!
That's right! You'll calculate cยฒ = 7ยฒ + 10ยฒ - 2(7)(10)cos(60ยฐ). What does it simplify to?
cยฒ = 49 + 100 - 70; so cยฒ = 79.
Well done! So c = โ79. Keep practicing this law; it's very powerful!
Law of Sines
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Now, letโs switch gears to the Law of Sines. This law is useful for finding unknown angles when you know one angle and its opposite side. Who can state this law for us?
Itโs a/sin(A) = b/sin(B) = c/sin(C)!
Correct! If we have triangle ABC where A = 45ยฐ, a = 10, and we need to find angle B, how would we set this up using the law?
We can set up the ratio as 10/sin(45ยฐ) = b/sin(B)! I think we need b to solve for angle B.
Right again. If we know other sides or angles, we can continue from there. Remember the ratios! They help in doing this without much complexity.
This is fun, using trigonometry together with geometry!
Constructing Circles
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Next, we're going to practice constructing circumscribed and inscribed circles around a triangle. Who can explain how we start constructing the circumcircle?
We need to find the perpendicular bisectors of two sides and then their intersection point!
Exactly! Once we find that intersection, we can set our compass to the distance from the center to any vertex to draw the circumcircle. And how do we find the incenter for the incircle?
We use the angle bisectors of the triangle's angles!
Great! Now letโs practice this together on paper and see how accurate we can be in our constructions!
Concurrency of Medians
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Letโs conclude our lessons by proving how the medians of a triangle meet at a single point called the centroid. Can anyone remember how we can prove this using coordinates?
We can assign coordinates to the vertices of the triangle!
You're correct! By using the coordinates of each vertex and employing the midpoint formula, we'll show how the three medians intersect at a point. Who wants to try this proof with me?
I will! Let's assign A(0,0), B(2,0), and C(1,3) for our triangle.
Perfect! Now compute the midpoints of AB, BC, and CA and then write the equations of the medians. Who can continue from here?
Introduction & Overview
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Quick Overview
Standard
The section elaborates on how to apply the Law of Sines and the Law of Cosines to solve for unknown sides and angles of triangles. Additionally, it discusses methods for constructing circumcircles and incircles, along with a proof regarding the concurrency of triangle medians at the centroid.
Detailed
Detailed Worked Examples
This section provides intricate and detailed worked examples illustrating the application of the Law of Sines and the Law of Cosines. These laws are essential for determining unknown side lengths and angles in various triangle configurations, particularly in non-right triangles. The examples explore the practical steps in solving triangles using these laws, which are crucial for understanding geometric properties and relationships.
Additionally, the section includes methods for constructing both circumcircles and incircles, offering hands-on experience with triangle properties. Lastly, we demonstrate the concurrency of the medians at the centroid, providing a proof through either coordinate methods or area techniques. Understanding these examples solidifies the theoretical knowledge of triangle properties and enhances problem-solving skills in geometry.
Audio Book
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Applying Law of Cosines to Find Unknown Side
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- Applying Law of Cosines to find unknown side
Detailed Explanation
In this example, we apply the Law of Cosines, which helps us determine the length of an unknown side in a triangle when we know the lengths of the other two sides and the angle between them. The Law of Cosines states that for any triangle, if a, b, and c are the lengths of the sides opposite to angles A, B, and C respectively, then:
aยฒ = bยฒ + cยฒ - 2bc cos(A).
By rearranging this formula based on the values we have, we can solve for the unknown side.
Examples & Analogies
Imagine you're a surveyor trying to determine the distance between two locations that form a triangle with a known angle. By applying the Law of Cosines, you can calculate that unknown distance just like triangulating a position on a map using landmarks.
Using Law of Sines in SSA Case
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- Using Law of Sines in SSA case
Detailed Explanation
The Law of Sines is incredibly useful when solving for unknown sides or angles in cases where you know two sides and an angle that is not included between them (SSA). The Law of Sines formula is as follows:
a/sin(A) = b/sin(B) = c/sin(C).
This means that the ratio of the length of a side to the sine of its opposite angle is constant for all sides of the triangle, allowing you to find unknown angles or sides with some rearranging of the formula.
Examples & Analogies
Think of it like using a pair of binoculars to spot a boat out on the water. If you know the angle to the boat and two distances (from yourself to two other known points), the Law of Sines helps you determine the distance to the boat, much like using triangulation to locate it on a map.
Constructing Circum- and Incircles
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- Constructing circumโ and incircles with compass/straightedge
Detailed Explanation
Circumcircles and incircles are important in triangle geometry. A circumcircle is the circle that passes through all the vertices of a triangle, while an incircle touches all the sides of the triangle. To construct these circles:
- For the circumcircle, find the perpendicular bisectors of at least two sides of the triangle. The intersection point of these bisectors gives you the center of the circumcircle. The radius can be measured from this center to any vertex.
- For the incircle, construct the angle bisectors of the triangle, where they intersect gives the incenter (the center of the incircle). The radius can then be determined by measuring the distance from the incenter to any side.
Examples & Analogies
Imagine you are designing a pizza and want to put toppings evenly over it. The circumcircle is like the outer crust edge, ensuring your toppings cover the entire pizza. The incircle is like the area within the crust that you want to top deliciously, ensuring no toppings spill over the edges.
Proof: Concurrency of Medians at Centroid
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- Proof: concurrency of medians at centroid (via coordinates or area method)
Detailed Explanation
Medians of a triangle are segments that connect each vertex to the midpoint of the opposite side. The centroid is the point where all three medians intersect. To prove that these medians are concurrent (i.e., they meet at one point), you can either use a coordinate proof or an area argument. In a coordinate proof, you can assign coordinates to the triangle's vertices and demonstrate that the equations of the medians give the same intersection point. For area, you can show that the triangle is divided into smaller areas that balance at the centroid.
Examples & Analogies
Picture a seesaw with three kids sitting at different points. The centroid acts like the balance point. Each median is a plank extending from the kids to the balancing point. Proving that all planks (medians) meet at the same spot explains why the seesaw balances. Each kid's weight creates a unique area contribution, ensuring equilibrium at the centroid.
Definitions & Key Concepts
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Key Concepts
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Law of Cosines: Allows solving for unknown side lengths when knowing two sides and the included angle.
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Law of Sines: Useful for finding unknown angles given one angle and its opposite side.
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Circumcircle: The unique circle that touches all vertices of a triangle.
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Incircle: The largest circle that fits entirely within a triangle, touching all sides.
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Centroid: The intersection point of the medians of a triangle.
Examples & Real-Life Applications
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Examples
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Using the Law of Cosines to find a side length when two sides and an included angle are given.
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Applying the Law of Sines to find an unknown angle of a triangle with one known angle and its opposite side.
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Constructing the circumcircle of a triangle using the perpendicular bisector method.
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Finding the centroid of a triangle given the coordinates of its vertices.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
๐ต Rhymes Time
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Cosines help with angles wide, find unknown sides and take pride.
๐ Fascinating Stories
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In a beautiful kingdom of triangles, the wise mathematician taught how all sides relate through special laws, guiding youth to construct circles that danced around vertices.
๐ง Other Memory Gems
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C for Cosine, A for Angle, then youโll dance with sides, itโs a triangle tangle!
๐ฏ Super Acronyms
CASH
- Cosine for Angles
- Sine for Heights.
Flash Cards
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Glossary of Terms
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Term: Law of Cosines
Definition:
A formula used to find an unknown side of a triangle when two sides and the included angle are known.
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Term: Law of Sines
Definition:
A formula relating the lengths of the sides of a triangle to the sines of its angles.
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Term: Circumcircle
Definition:
The circle that passes through all the vertices of a triangle.
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Term: Incircle
Definition:
The circle inscribed within a triangle, tangent to all three sides.
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Term: Centroid
Definition:
The point where the medians of a triangle intersect.