Geometry (Proofs)

Question 1

Describe Garfields Proof of the Pythagorean Theorem

Answer 1

1. Construct a right triangle with Length “B,” Height “A,” and Hypotenuse “C.”
2. Create a Congruent Triangle (Stacked on Top) creating a side “A + B” as the height.
3. Label the Angles and Prove that the missing Angle = 90°.
4. Create an equation using the Area of a Trapezoid,

Height x ((Top Base + Bottom Base)/ 2)

5. Create an equation using the Area of the Triangles,

(Base x Height) / 2 and simplify both as a single equation.

Question 2

Create a Proof for the 45°-45°-90° Triangle Equation

Answer 2

1. Establish that a 45°-45°-90° Triangle has two equal length sides.
2. From that information we can Establish that in the Pythagorean Theorem, A = B, and we can Substitute one with the other.
3. Simplify the equation and solve for A or B to determine that

A = B = (√2/2) C

or

(√2) A = (√2) B = C

Question 3

Create a proof for the sides of a 30° 60° 90° Triangle.

Answer 3

1. Create an Equilateral Triangle and label all the sides and angles.
2. Split the Equilateral Triangle down the middle and label the new sides and angles.
3. Create an equation using the Pythagorean Theorem to find the missing side.
4. Solve and Simplify the equation for the missing side “a.”

Question 4

Create a proof for the Law of Sines

Answer 4

1. Create a triangle that doesn’t have a 90°
2. Label two sides and angles opposite each other.
3. Drop an altitude through the middle of the two sides and angles.
4. Create two equation using the two angles as a function of Sine.
5. Solve and simplify the two equations as a single equation for the new side “x”.

Question 5

Explain how you get the Arc Length (Degrees) Equation

Answer 5

1. You need the central angle that subtends the arc length.
2. From there, you can divide that angle (In degrees) by (360°) to receive the ratio of the Arc Length in comparison to the entire Circumference.
3. Then, you multiply the ratio of the Arc Length and the Circumference to receive the total Arc Length

* From that information, you can create and manipulate the equation to determine the Arc Measure or Circumference of a Circle (In degrees)

Question 6

Explain how you get the Equation to convert (Radians to Degrees)

Answer 6

360 (Degrees) = Circumference

2π (Radians) = Circumference

360 (Degrees) = 2π (Radians)

180 (Degrees) = π (Radians)

180/π (Degrees) = (Radians)

Question 7

Explain how you get the Arc Length Equation in Radians.

Answer 7

1. We established the Arc Length (Degree Equation) to be

Arc Length = (Central Angle/ 360°) (Circumference)

2. We can substitue (360-Degrees) with (2π-Radians).
3. We also substitute the (Circumference) with (2πr)
4. The new equation we get is

Arc Length = (Central angle/ 2π) (2πr)

5. Then we simplify (Central angle x 2πr) / (2π) to get

Arc Length = Central Angle x Radius

Question 8

Create a Proof that Triangles created by angles subtending a circles diameter are Right Triangles.

Answer 8

1. Create an image of a random triangle subtending a circles diameter.
2. Cut the triangle in half and label the sides as “r”
3. Realize that the central angle is twice as long as the other angle subtending the same arc and label the angles.
4. Create an equation for the central triangle to determine the other two angles.
5. Once solved, add the theata to the other angle to create a 90° angle