/2 Pythagorean Theorem
One important application of square roots has to do with right triangles. Do your remember what a right triangle is. The RIGHT TRIANGLE is a triangle where one of the angles is a RIGHT ANGLE or one that measures 90 degrees.
The hypotenuse of the right triangle is the side that is opposite of the right angle. The right angle
is symbolized by the image "Figure (2).
In the figure (1) above, side "c" is opposite the "right angle" that is shown in the "right triangle".
CLICK FIG (3) FOR VIDEO
CLICK FIG (4) TO VIEW
a² + b² = c² . . . In previous lessons we learned about finding the area of squares. The reason that these lessons PRECEDED this current lesson on the Pythagorean Theorem is because when we use the Pythagorean Theorem we are actually finding the area of the missing squares shown in figure (3).
The AREA of any shape is simply "the size" of the surface. In fig (4) you will notice that the AREA of each shape is 9 square units.
Notice the square shape in this figure, it can be described for us as having three blocks in the horizontal direction (sideways/back and forth) {1,2,3}. It can also be described for us as having three blocks in the vertical direction (up and down) {1,4,7}.
The AREA or the SIZE of this square is determined by simply counting the squares that are contained inside the square. When we do this, we count a total of NINE squares and we write this down as 9 square inches, or 9 square cm . . .
In this particular instance we counted NINE squares within the square. We can also use MULTIPLICATION to do the same thing. Supposing the square were much larger (for instance the outside wall of a block building), rather than counting each individual block that is used in the wall, we could could count the blocks along the lower edge and also the number of blocks along ONE outside edge as shown to us in figure (5).
Use your imagination here, the number 3 along the left side of the image can represent 3 blocks, 3 feet, 30 feet or 300 feet. The same is true of the number 5 along the top, it could represent 5 block, 5 feet, 50 feet or 500 feet - you get the idea.
The FORMULA for finding the area {the size} of a FLAT surface is always the same. Either each individual block or square on the shape is counted OR the length of one side is multiplied by the length of a perpendicular side as shown.
Now that we have reviewed the concept of AREA, lets recall our knowledge of SQUARE ROOTS.
At this point in our lesson - when we mention SQUARE ROOTS, we should not be introducing an unfamiliar concept. In the figure (5) above, the square unit area of the rectangle is 3 x 5 or 15 square units. Conversely if we work BACKWARDS from a given surface area, how do we find the respective SIDES of the shape that resulted in our 15 square unit size?
I wanted to use this 3 x 5 rectangle to illustrate the point that when we mention the Pythagorean Theorem, we are only talking about SQUARES not rectangles.
If we take the SQUARE ROOT of 15, the answer is ≈ 3.87. What this means is that a SQUARE with each of it's four sides measuring ≈ 3.87 will produce an AREA of 15 square units of measure.
You will note however that the shape in figure (5) is NOT a square and the sides shown are NOT ≈ 3.87 each but rather 3 and 5 respectively . . . consequently the PYTHAGOREAN THEOREM does not apply to any shape other than a SQUARE.
Notice figure (6). See how we get the term "3 squared"? In this instance 3 square units x 3 square units = 9 square units.
BACKWARDS to determine the length of the width and the height of the square, we would change directions and simply use the concept of SQUARE ROOTS.
NOW our knowledge of ALGEBRA comes in handy here. By this I refer to how we can solve for the missing value in the equation given as a² + b² = c².
If we know the value of any two of these three variables, then using ALGEBRA we can "solve" for the third variable that is missing.
But before we begin to use our knowledge of ALGEBRA to solve for the missing variable (the side of one of the RIGHT triangles), we will have a brief review of "how to find X" in a standard ALGEBRAIC equation because this will be exactly what we will be doing with the Pythagorean Theorem.
Think about the word ALGEBRA as really being a PUZZLE . . . for instance: if we have two apples how many more apples do we need before we have a total of four? Algebra really is that simple!
____ + 2 = 4 We usually see this type of equation written as follows with the instruction given as FIND X in the equation:
The solution is found using ALGEBRA . . . if one apple weights 1/2 pound and if you have two apples . . . then the solution is found by substituting 1/2 pound for each apple symbol in figure (8).
!/2 lb + 1/2 lb = 1 lb . . . ALGEBRA is all about changing, interchanging and in some cases INTERCHANGING SYMBOLS with numbers. As we proceed in to higher levels of ALGEBRA, the problems do get a bit more complex but the concept remains as simple as described in this review.
Now we are ready for the Pythagorean Theorem! Which uses the concept of AREA to solve for the unknown lengths of the sides of a RIGHT TRIANGLE.
Do you know what the term "hypotenuse" means? In words the hypotenuse is the side of a triangle that is opposite of the right angle in a RIGHT ANGLED triangle.
Sample Questions: Only one of these triangles is really a right triangle. Which one is it?
FOLLOW THIS LINK FOR PRACTICE PROBLEMS
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| The Pythagorean Theorem Figure (1) |
One important application of square roots has to do with right triangles. Do your remember what a right triangle is. The RIGHT TRIANGLE is a triangle where one of the angles is a RIGHT ANGLE or one that measures 90 degrees.
The hypotenuse of the right triangle is the side that is opposite of the right angle. The right angle
 |
| Fig (2) |
In the figure (1) above, side "c" is opposite the "right angle" that is shown in the "right triangle".
CLICK FIG (3) FOR VIDEO
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| Figure (3) |
When a triangle has a right angle (90°) . . . and if the squares are made on each of the three sides as shown . . . then the biggest square (the red one) has the EXACT SAME AREA as the "SUM" of the other two.
What this means is that if we find the area of the green square "a" and add this to the area of the purple square "b" then the result will be the area of the red square "c".
So figure (3) is a visual definition (representation) of the Pythagorean Theorem . . . while the following is the non-visual representation of the same thing.
CLICK FIG (4) TO VIEW
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| Example of AREA fig (4) |
The AREA of any shape is simply "the size" of the surface. In fig (4) you will notice that the AREA of each shape is 9 square units.
Notice the square shape in this figure, it can be described for us as having three blocks in the horizontal direction (sideways/back and forth) {1,2,3}. It can also be described for us as having three blocks in the vertical direction (up and down) {1,4,7}.
The AREA or the SIZE of this square is determined by simply counting the squares that are contained inside the square. When we do this, we count a total of NINE squares and we write this down as 9 square inches, or 9 square cm . . .
In this particular instance we counted NINE squares within the square. We can also use MULTIPLICATION to do the same thing. Supposing the square were much larger (for instance the outside wall of a block building), rather than counting each individual block that is used in the wall, we could could count the blocks along the lower edge and also the number of blocks along ONE outside edge as shown to us in figure (5).
 |
| Figure (5) |
The FORMULA for finding the area {the size} of a FLAT surface is always the same. Either each individual block or square on the shape is counted OR the length of one side is multiplied by the length of a perpendicular side as shown.
Now that we have reviewed the concept of AREA, lets recall our knowledge of SQUARE ROOTS.
At this point in our lesson - when we mention SQUARE ROOTS, we should not be introducing an unfamiliar concept. In the figure (5) above, the square unit area of the rectangle is 3 x 5 or 15 square units. Conversely if we work BACKWARDS from a given surface area, how do we find the respective SIDES of the shape that resulted in our 15 square unit size?
I wanted to use this 3 x 5 rectangle to illustrate the point that when we mention the Pythagorean Theorem, we are only talking about SQUARES not rectangles.
If we take the SQUARE ROOT of 15, the answer is ≈ 3.87. What this means is that a SQUARE with each of it's four sides measuring ≈ 3.87 will produce an AREA of 15 square units of measure.
You will note however that the shape in figure (5) is NOT a square and the sides shown are NOT ≈ 3.87 each but rather 3 and 5 respectively . . . consequently the PYTHAGOREAN THEOREM does not apply to any shape other than a SQUARE.
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| Figure (6) |
We could also say 3² = 9. Now if we started with 9 square units that represented the SIZE of the square in figure (6) and wanted to work
BACKWARDS to determine the length of the width and the height of the square, we would change directions and simply use the concept of SQUARE ROOTS. |
| CLICK IMAGE FOR VIDEO LESSON ON SQUARE ROOTS |
NOW our knowledge of ALGEBRA comes in handy here. By this I refer to how we can solve for the missing value in the equation given as a² + b² = c².
If we know the value of any two of these three variables, then using ALGEBRA we can "solve" for the third variable that is missing.
 |
| Fig (7) |
But before we begin to use our knowledge of ALGEBRA to solve for the missing variable (the side of one of the RIGHT triangles), we will have a brief review of "how to find X" in a standard ALGEBRAIC equation because this will be exactly what we will be doing with the Pythagorean Theorem.
Think about the word ALGEBRA as really being a PUZZLE . . . for instance: if we have two apples how many more apples do we need before we have a total of four? Algebra really is that simple!
|
| Fig (7) |
X + 2 = 4 which actually means . . . if we have a total of 4 apples, how many more apples must ADD to the two that we already have to make the equation true?
Another example might be like this: One apple added to another apple results in a total of 2 apples. In equation form this would be written as
We may already know that we have a total of two apples. We might then be asked, "If one apple weights 1/2 pound, how much will two apples weight?
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| fig (8) |
!/2 lb + 1/2 lb = 1 lb . . . ALGEBRA is all about changing, interchanging and in some cases INTERCHANGING SYMBOLS with numbers. As we proceed in to higher levels of ALGEBRA, the problems do get a bit more complex but the concept remains as simple as described in this review.
Now we are ready for the Pythagorean Theorem! Which uses the concept of AREA to solve for the unknown lengths of the sides of a RIGHT TRIANGLE.
Do you know what the term "hypotenuse" means? In words the hypotenuse is the side of a triangle that is opposite of the right angle in a RIGHT ANGLED triangle.
Notice that all three sides of this triangle are named. Notice the Greek symbol "θ" . . . this symbol represents an angle. The idea is that the length of the side that is opposite the value of this angle is on one side of the 90° while the length of the side that is adjacent to the value of this angle is on the other side of the 90°.
Go back to the beginning of this lesson and review figure (3). The three squares are all of different sizes. The lengths of the four sides of the green square are all equal to one another. The lengths of the four sides of the purple square are all equal to one another. The lengths of the four sides of the red square are all equal to one another.
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| Figure (3) |
Since the four SIDES of each one of the respective squares are equal to "each other" (but NOT equal to the lengths of the sides of the OTHER squares) . . . then one side of each square represents ONE LEG of the right triangle shown in fig (3).
a = to the length of all four sides of the green square.
b = to the length of all four sides of the purple square.
c = to the length of all four sides of the red square.
The calculated area of square a ADDED to the calculated area of square b is equal in value to the calculated area of square c. This is what a² + b² = c² means.
Sample Questions: Only one of these triangles is really a right triangle. Which one is it?
![[image]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vN-rReFoIrAD0d8RoX0gzBL5xoDPhupJT5-HnrB4JkxZyLw-HupBFnuSIVCzRH2lHcWjOiIVgtdtHka-8CcD6s9OZ-l71D-l7dU6iRSnKDGrr76PjBtnKxKP3XIJbJxUwSs3DaOoN012aw8ecZXf_jB3vfwto=s0-d) |
| Sample problem (1) |
![[image]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_svpSslkDGz7Kd0dU29EQExyEMv2QWb4itiuxc-NpNcg_5i1EN75yrJwp0B47-ttUiswv9t87Z2JnUi05MAdMXGHot-0CPTwscg_6VOyzTE6AEMbfWpl1hpqcOxVqzjtiiSPJdgdaAWi2Un-Lv4cBIFWIlvrdw=s0-d)
A scalene obtuse angled triangle
A scalene acute angled triangle
A scalene right angled triangle
FOLLOW THIS LINK FOR PRACTICE PROBLEMS
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