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Home , Base (geometry)

Tetrahedron By Group 4 Group 4

Rachel Haynes

Shayaan Chaudhary

Amy Wu The Problem Overall Plan

- Find the of the base of the - Find the height of the tetrahedron - Use base*height/3 to find the volume Definition of

Definition: A point on a segment that divides it into two equal parts.

Application: Midsegment Theorem

Theorem: The segment in a connecting the of two sides of the triangle is parallel to and half the of the third side.

Application: Midsegment Theorem Proof Finding the Area of the Base

Strategy 1: Heron’s Formula

Strategy 2: Law of Cosines Strategy 1: Heron’s Formula

Formula:

Application: Heron’s Formula Proof (Attempt) Strategy 2: Law of Cosines

∠CQR

110.25 = 130.25 - 110cosQ cosQ = .181818 ∠BQP Q = 79.5243 degrees

∠PQR

∠CQR + ∠BQP + ∠PQR = 180 degrees 30.25 = 210.25 - 210cosc 79.5243 + 31.0027 + ∠PQR = 180 degrees Cosc = 31.0027 degrees ∠PQR = 69.473 degrees Strategy 2: Law of Cosines

10.5 ∠PQR = 69.473 degrees P Q h 10 5.5

R Law of Cosines Proof

A

b a h

B C x c - x c Finding the Height

Strategy 1

Strategy 2 Strategy 1 New Right Triangle Making an Assumption Sine Definition

Definition: The trigonometric function that in a right triangle is equal to the ratio of the side opposite of a certain to the hypotenuse of the triangle.

Application: Strategy 2: Graphing Method

Outline: - Graph the base of the tetrahedron on a 2-D XY plane - Find the coordinates of all three vertices - Set up a system of equations involving 3 dimensional distance formulas to find the fourth (the point where A,B,C coincide at the top of the tetrahedron - Use the Z-coordinate of that vertex as the height of the tetrahedron Graphing the base 1. Draw a cartesian plane and draw side PR of the base on the x-axis with point P on the origin 2. Draw point Q arbitrarily above the x-axis and connect all the points 3. Label all the side (determined by midsegment theorem on slide 6) Finding The Coordinates of the Vertices

- Since Point P is on the origin, the coordinates can be assumed to be (0,0) - Point R is on the x-axis and so the y-coordinate is 0. The x-coordinate can be seen to be 10 as the horizontal distance between P and R is 10. - Point Q can be found using trigonometry on the next slide. Finding the Coordinates of the Vertices (cont.)

Point Q:

- Use Law of Cosines formula to find Angle P: - Use the three side lengths labeled with side c as side QR, and use angle C as angle P: - Draw an from Q to the base and label it point D Finding the Coordinates of the Vertices (cont.)

- Use trigonometric ratio to find length of segment QD - Use sin P = opposite/hypotenuse - Substitute appropriate values - QD = 5.4833 - Use Pythagorean Theorem to find PD - a^2 + b^2 = c^2 - Substitute appropriate values - PQ^2 - QD^2 = PD^2 - PD = 9 - Since QD is the distance of point Q from the x-axis, QD is the y-value of the coordinate - Since PD is the distance of point Q from the y-axis, PD is the x-value of the coordinate - Therefore the coordinates of point Q are (9,5.4833) Set Up System of Equations

- Now imagine the top vertex of the tetrahedron, as above this 2D shape on a third dimension of height. Mark it as point E. This now makes it an XYZ coordinate graph. - Since points P, Q, and R are all coplanar on the xy-plane, their “Z” coordinate is 0. - The distance formula of a 3D distance is: where a variable with the subscript 2 represents the coordinates of point E and variables with the subscript 1 represent the coordinates of one of the initial vertices just found. - Write a distance formula for each distance from the three points on the base: PE, QE, and RE - Shown on next slide Set Up System of Equations (cont.) Set Up System of Equations (cont.) - We know that when the three sides of triangle ABC were folded upwards, the sides of each of the outer (BQP, ARP, CQR), became the three edges of the tetrahedron which led to point E. - Therefore you can set each distance formula equal to the side length which it corresponds to: - PE: 11/2 - QE: 10 - RE: 21/2 - Now, if you set the corresponding distance formulas equal to the side lengths, you have a system of three equations which you can solve: Use Z-coordinate as height of tetrahedron

- Since the z-coordinate represents the 3-dimensional height of the graph, the z-coordinate can be seen as the height of the tetrahedron. - Now this height can be used in next step to find the volume of the tetrahedron. - Z = 4.9923 = h Final Volume

USING HEIGHT OF STRATEGY 1

USING HEIGHT OF STRATEGY 2 Conclusion

- We also tried another solution which used the orthocenter and a right triangle formed between P, the orthocenter, and the top vertex of the tetrahedron, however we could not come up with a final proof which came out to exactly 45, but rather 46.54. - One generalization we considered was the distance formula as it is an extension of the pythagorean theorem. Thank you