College Prep Math
Shapes and Measurements

Basics

B.1) The greatest all-time measuring tool:
The Pythagorean theorem says that when you go with a right triangle, then you are guaranteed that:
The square of the length of the long side is the sum of the squares of the lengths of the short sides.
This guarantees that the distance  between a point {x,y} and another point {a,b} is

B.1.a.i) The Pythagorean theorem says that when you go with a right triangle, then you are guaranteed that:
The square of the length of the long side is the sum of the squares of the lengths of the short sides.

Look at this triangle:

Folks call this a right triangle because the two shorter sides are perpendicular to each other.

In other words, the angle at • is a right angle.

What does the the Pythagorean theorem tell you about how the lengths of the three sides  of a right triangle relate to each other?

Answer:

Take another look at a right triangle:

The Pythagorean theorem says that when you go with a right triangle, then you are guaranteed that:
The square of the length of the long side is the sum of the squares of the lengths of the short sides.

Some folks like to call the long side of a right triangle by the old-fashioned name hypotenuse.
This term was derived from the ancient Greek word hypatos which means "the greatest."
For this course,hypotenuse has been translated into modern American.

In the colors of this graphic, the Pythagorean theorem says

             .

If you become curious about the reasoning underpinning the Pythagorean theorem, visit the next Basic problem.

B.1.a.ii) The Pythagorean theorem guarantees that the distance  between a point {x,y} and {0,0} is

Here's a random point {x,y} shown with {0,0}:

That gold point •  sits at {0,0}. That blue point •.sits at {x,y}.

Measure the distance between the two points.

Answer:

This is a job for a right triangle and the Pythagorean theorem.

Throw in the point {x,0} • on the x-axis directly below {x,y} •.

See the right triangle emerge:

The distance between {x,y} and {0,0} is the length of the long side of this right triangle.
  Label the lengths of the short sides:

Review:

Animate by double clicking one of the plots.

The distance between {0,0} and {x,y}  is the length of the long side of this right triangle.
The Pythagorean theorem  guarantees that the square of the length of the long side is the sum of the squares of the lengths of the short sides.

In the colors of the graphic :.

In other words,
                    = +

.

So the distance between {0,0} and {x,y} is

Wrap it up:

B.1.a.iii) The Pythagorean theorem guarantees that the distance  between a point {x,y} and another point  {a,b} is

Here are two points {x,y} •  and {a,b} • :

That gold point •  sits at {a,b}. That blue point • sits at {x,y}.

Measure the distance between the two points.

Answer:

This is another job for a right triangle and the Pythagorean theorem:
Throw in this new point {x,b} •  directly to the right of {a,b} •  and directly underneath {x,y} • :

See the right triangle emerge:

You are guaranteed that this is a right triangle with the right angle at • because
one of the short sides •• is parallel to the x-axis and the other •• is parallel to the y-axis

Review:

Label the lengths of the short sides:

Review:

Animate by double clicking one of the plots.

The distance between {a,b} and {x,y}  is the length of the long side of this right triangle.
The Pythagorean theorem  guarantees that the square of the length of the long side is the sum of the squares of the lengths of the short sides.

In the colors of the graphic :.

In other words,
                    = +

.

So the distance between {a,b} and {x,y} is

Wrap it up: