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A short class with Sarah and her students. The other ways are under More.
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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 a² = b² + c² - 2bc cos(A)?
Exactly! This formula connects the lengths of the sides with the cosine of the included angle. Let’s consider an example: suppose we have a triangle with sides a = 7, b = 10, and the angle A = 60°. What will be the length of side c?
We can plug those values into the formula!
That's right! You'll calculate c² = 7² + 10² - 2(7)(10)cos(60°). What does it simplify to?
c² = 49 + 100 - 70; so c² = 79.
Well done! So c = √79. Keep practicing this law; it's very powerful!
Now, let’s 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?
It’s a/sin(A) = b/sin(B) = c/sin(C)!
Correct! If we have triangle ABC where A = 45°, 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(45°) = 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!
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 let’s practice this together on paper and see how accurate we can be in our constructions!
Let’s 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?
Allows solving for unknown side lengths when knowing two sides and the included angle.
Useful for finding unknown angles given one angle and its opposite side.
The unique circle that touches all vertices of a triangle.
The largest circle that fits entirely within a triangle, touching all sides.
The intersection point of the medians of a triangle.
Using the Law of Cosines to find a side length when two sides and an included angle are given.
Applying the Law of Sines to find an unknown angle of a triangle with one known angle and its opposite side.
Constructing the circumcircle of a triangle using the perpendicular bisector method.
Finding the centroid of a triangle given the coordinates of its vertices.
Cosines help with angles wide, find unknown sides and take pride.
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.
C for Cosine, A for Angle, then you’ll dance with sides, it’s a triangle tangle!
CASH: Cosine for Angles, Sine for Heights.
Term
Law of Cosines
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Circumcircle Construction
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Incenter
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Centroid Location
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A formula used to find an unknown side of a triangle when two sides and the included angle are known.
A formula relating the lengths of the sides of a triangle to the sines of its angles.
The circle that passes through all the vertices of a triangle.
The circle inscribed within a triangle, tangent to all three sides.
The point where the medians of a triangle intersect.
This section presents detailed worked examples utilizing the Law of Sines and Law of Cosines, showcasing practical applications of triangle properties.
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.
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.