20170227

Author:: Adam

Problem 1

In the right triangle below, sides are marked with lengths a, b, c (around the outside of the larger triangle), and two smaller segments, r, s compose the side of length c. The height we call h. a) If a = 4 and b = 6 find all other side lengths. b) If a = 4 and c = 10, it should be obvious how you find all sides of the triangle, so don't bother doing it. Just make sure you see how the solution to part (a) will work the same here. Likewise if you know b and c. c) If a = 2 and r = 1 find all other side lengths. d) If a = 3 and s = 3 find all other side lengths. This is relatively hard, so as a hint, use the facts

Problem 2

Let's come up with an alternate version of the Pythagorean theorem. Rather than using the areas of squares on the sides, like so

we can use triangles like so

Show that the area of

plus the area of

is equal to the area of

Problem 3.

In the diagram below

. Show that

has the same area as

Problem 4.

In the diagram below, d is the length of

and this is the "distance" from A to the line. The distance to the line is the length of the line segment that starts at A, ends at the line, and forms a right angle with the line. Point C is some other point on the line. If AC = 20 and

find the distance from A to

Problem 5.

The following object is called a "vector". It is a directed line segment, and the direction is indicated by the direction of the arrow head at the end of it, and it's one of the most important and fundamental mathematical tools for analyzing Physics and some other Sciences. It is defined by a pair of coordinates, in this particular case, the vector has coordinates <4,4>. This just means the vector is drawn by moving four units right and four units up from its starting location. The "magnitude" of the vector is defined as the length of the line segment. Find the magnitude of this vector.

Problem 6.

The figure below is a cube. The surface area of square

is 100. Find the length of the line segment

Problem 7.

Suppose the ratio of a leg to the hypotenuse of a right triangle is 2:5. If one leg of the triangle is of length 10, find two possible values for perimeter of the triangle.

Problem 8.

Suppose that a side length of an equilateral triangle is 10. On each edge, draw its midpoint. Connect all of these midpoints and find the area of the triangle formed inside. Also prove that this cuts the triangle into four congruent triangles.