45-45-90 Right Triangle

Congruent Triangles


Monday's Review


1

The total distance from B to C to D is given by the equation

prob1pic.png


Graphing this, we look for the minimum value on the graph. The ordered pair will have the form (x,shortest distance). Look at the graph above.

2


prob2pic.png
Begin by sketching a graph of the situation. Then apply the Pythagorean Theorem to the isosceles triangle (or two right triangles) in the figure.

The coordinates are show.

Exactly, the coordinates are




YOU SHOULD BE ABLE TO DO THIS PROBLEM WITHOUT A CALCULATOR!

3

The midpoint is (-3,6). The slope of the segment is -1. The slope of the perpendicular bisector is +1. Using point-slope form, the equation of the perpendicular bisector is y-6=1(x+3). In slope-intercept form is y=x+9.

4

This is my hexagon. I am sure there are other ways to draw this.

hexagon.png

5


Refer to the pencast of this problem on Page 4, #8.
Let d = distance of the race.



So the difference in their times must be 6d - 5d = 1, or d = 1 mile.

6




The perpendicular slope is



The equation of the line in point-slope form is



8

For this problem, I used that idea that we developed when we first encountered this problem, that the square of the length of the side of the middle square is equal to the product of the lengths of the sides of the outer two squares.
Therefore, there are three different equations that need to be solved for the three different cases:



leading to the three solutions




9

Using the Pythagrean Theorem,


or



The left side can be factored as a difference of two squares as (c+b)(c-b). The right side can be factored in the following ways: 1x400, 2x200, 4x100, 5x80, 8x50, 10x40, 16x25, and 20x20. Only the second, third, fifth, and sixth of these factor pairs can be used.

For example, (c+b)(c-b)=(2)(200) leads to the system c+b=2 and c-b=200. This means c=101 and b=99. Therefore, one triple is 20, 99, and 101.

You should confirm the following, but the 4x100 leads to the triple 20, 48, 54.
8x50 leads to 20, 21, 29.
10x40 leads to 15, 20, 25.

10

Recalling how the sides of an isosceles right triangle are related, an expression for the perimeter would be



which means that



11

The vector [8,-6] has a length of 10 units.
The vector [-8,6] points in the opposite direction.
Multiplying by the scalar 0.5, the vector [-4,3] points in the same direction but has a length of 5 units.
Multiplying by the scalar 5, the vector [-20,15] is the desired vector.