MAT1360 Classwork
MAT 2401 Homework 3.5 Name:________________________________
• Do not copy homework. You never learn if you do that.
• Do all calculations by hand unless otherwise stated. (You can check your answers with SageMath.)
• Put down the necessary details.
• Full sentence answers are expected for some problems.
Summary
Area of a Triangle
|The area of the triangle with vertices [pic], [pic], and [pic] is given by |
|[pic] or [pic] or [pic] |
|The textbook tries to avoid the notation nightmare by using the 3rd formula with the understanding that we will use the positive |
|answer as the area. |
Collinear
|3 points [pic], [pic], and [pic]are collinear if and only if [pic] |
Cramer’s Rule
|If the system [pic]has a unique solution, then |
|[pic], [pic], [pic] |
1. (5 points) Use Cramer's Rule to solve the system of linear equations.
[pic]
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2. (5 points) Find the area of the triangle with vertices [pic], [pic], [pic].
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3. The following is the diagram of a triangle with all angles less than [pic].
[pic]
(a) (2 points) Redraw the triangle above with the following added details.
(i) Draw a perpendicular line from B to the line segment AC.
(ii) AC is dived into 2 segments: AD and DC. Let the length of AD and DC be respectively [pic] and [pic].
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(b) (4 points) Use elementary geometric arguments to demonstrate that
[pic].
Be sure to state your arguments carefully.
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Similarly, we can draw perpendicular lines to the other two sides AB and BC of the triangle. Each time, we will get a linear equation in [pic], [pic], and [pic]. The three linear equations form the system
[pic].
Here we can consider [pic], [pic], and [pic] as constants with unknowns [pic][pic], and [pic]. To make things easier, let [pic], [pic], and [pic]. Then the system becomes
[pic].
(c) (4 points) Use the Cramer’s Rule to solve for [pic].
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(d) (2 points) Substitute [pic] into the answer of (c) and solve for [pic]. This is the Law of Cosines.
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[pic]
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