THE PYTHAGOREAN THEOREM

 

Pythagorean Theorem:  In any right triangle the sum of the squares of the legs is equal to the square of the hypotenuse

 

In other words, in any right triangle, if a and b are the sides adjacent to the right angle (the legs), and if c is the side opposite the right angle (the hypotenuse), it will always be true that

 

 

What this implies is that if you know the lengths of the sides of a triangle you can determine whether it is a right triangle, and if you know that a triangle is a right triangle, then given the length of two of the sides you can calculate the length of the third side.

 

The triangle on the right is a right triangle. Therefore .

Check:  9 + 16 = 25.

 

Pythagorean Triples:  Sets of integers that satisfy the Pythagorean Theorem.

For example, the set {3, 4, 5} is a Pythagorean triple, and any multiple of this set is also a Pythagorean triple.

 

Example:  Find the hypotenuse of a right triangle if the lengths of the legs are 6 and 8.

 

Solution:

Let c be the hypotenuse of the triangle.

Then

10 = c

 

Test hint:  Finding the value of c requires too much calculation. 

Recognize that the set {6, 8, 10} is a multiple of the set {3, 4, 5}.  Therefore, c = 10.

 

Example:  If one leg of a right triangle is 5 and the hypotenuse is 13, find the other leg.

 

Solution: 

Let b be the other leg.

Then

b = 12

 

 

 

Test hint:  The second most common Pythagorean triple is {5, 12, 13}.

 

A right triangle whose legs have equal length is an isosceles right triangle, and the angles opposite the two equal sides are 45° angles.

 

You might encounter isosceles right triangles when looking for the diagonal of a square, for example. 

Note that

 

 

If one of the equal sides of an isosceles right triangle has length s, then the hypotenuse is .

 

Another commonly encountered right triangle is the 30°–60°–90° triangle.

 

If the hypotenuse is 2, the side opposite the 30 degree angle is half the hypotenuse, or 1, and the side opposite the 60 degree angle is .

Note that

 

If the hypotenuse is s units, the side opposite the 30° angle is  units and the side opposite the 60° angle is  units.

 

The Common Pythagorean Triangles

 

Know these!!!

 

Your Basic 3–4–5 Triangle

 

3

 

6

9

4.5  (=3 x 1.5)

4

 

8

12

6

5

 

10

15

7.5

 

 

The Common 5–12­–13 Triangle

 

5

 

10

12

 

24

13

 

26

 

 

Isosceles Right Triangle

 

1

 

s

8

1

 

s

8

 

s

8

 

 

The 30°-60°-90° Triangle

 

1

 

 

10

 

 

10

2

 

1

 

 

20

 

Some Other Triples You Might Never See

 

8

7

15

24

17

25

 

 

Example:  Find the height of a rectangle whose length is 3 inches and whose diagonal is 4 inches.

 

Answer:

Caution:  This was a trap!  Just because a right triangle has sides with lengths 3 and 4 don't assume it must be a 3-4-5 right triangle.

Draw a picture.  Note that 4 is the hypotenuse, not one of the legs. 

 

Let a be the height of the triangle.  Since it is a right triangle we know that

 

 Example:  If the triangle on the right is an equilateral triangle with side s and altitude h, find the height and the area in terms of s.

 

Answer: The height is , and the area is .

 

We can memorize the formula for the area of an equilateral triangle, but since we don't see it that often it's nice to know we can derive it.

 

A perpendicular bisector will create two  30-60-90 right triangles, which we can use to find the height and the area.

The height is

A =

= =

 

Example:  In the circle below, O is the center and B is a point on the circle, ABCO is a rectangle, OC = 4 and OA = 3.  Find the area of the circle.

 

 

Answer:  25

Solution:  Note that OB is the radius of the circle.  Since ABCO is a rectangle, angle BCO is a right angle and triangle BCO is a right triangle with legs of length 3 and 4.  By the Pythagorean Theorem, we know the hypotenuse (radius) is 5.  The area is =  = 25 .

 

 

Example:  Find the perimeter and the area of the figure below.

Let AB = 5, BC = DC = 8, and ED = 4, and ÐB = ÐC = ÐD = 90°.

Answer:  Perimeter = 30 units, area = 58 square units.

Solution:  We need to find the length of EA to find the perimeter.  We can extend the sides AB and ED to create a quadrilateral as pictured below.  If we can find the area of the quadrilateral and the area of the shaded triangle AFE, we can use this to find the area of ABCDE.

Extend the sides AB and ED to form quadrilateral FBCD.

Since angles B, C, and D are right angles, angle F must also be 90 degrees (the sum of the degree measures of a quadrilateral is 360 degrees).

Since two adjacent sides have equal length, FBCD is a square.

Therefore, FB = DC = 8 and FA = FB - AB = 8 - 5 = 3.

Similarly, FD = BC = 8 and FE  = FD - ED = 8 - 4 = 4.

Now we can use the Pythagorean Theorem on the shaded triangle AFE.  Since the legs have lengths 3 and 4 and since angle F is a right angle, the hypotenuse EA, has length 5. 

Therefore, Perimeter ABCDE = AB+BC+CD+DE+EA = 5 + 8 + 8 + 4 + 5 = 30.

 

Area of square FBCD = = 8 x 8 = 64.

Area of triangle AFE = bh = (3)(4) = 6.

Area of ABCDE = Area of square FBCD - Area of triangle AFE

= 64 - 6 = 58 square units.

Alternate solution:  We might also have sectioned ABCDE as below, where AF = 4 and EF = 5.  Again, if we apply the Pythagorean Theorem to triangle AFE we find EA = 5, and from this we can find the perimeter.

 

The areas of the sections are as indicated below.

Triangle AFE = bh = (3)(4) = 6.

The area of the upper rectangle is 4 x 5 = 20, and the area of the lower rectangle is 4 x 8 = 32.

 

 

Example:

Super Pythagorean Problem

 

If ABC, ADB and BDC are similar right triangles, find the lengths of x, y, and z.

 

Answer:  y = 3, x = 16/3 or 5.33, z = 20/3 or 6.67

 

Note that triangle BCD is a right triangle with sides 4 and 5.  Therefore, it is a 3-4-5 right triangle.  Now we can use the fact that similar triangle have sides in proportion to each other.  It helps to draw the triangles separately. Then it becomes easier to see that

3/4 = 4/x

3x = 4·4

x = 16/3 = 5.33

 

Finding z using Triangle ACB:

Now to find z we can use the Pythagorean Theorem.

52 + z2 = (x + y) 2

25 + z2 = (16/3 + 3) 2 - 25

z2 = (8.33)2 - 25

z2 = 69.44 - 25

z2 = 44.44

z = 6.67

 

Finding z using Triangle ADC:

z2 = 42 + (5.33) 2

z2 = 16 + 28.44

z2 = 44.44

z2 = 6.67

 

Final comment on Triangle ADC.  We can see that this is a 3-4-5 triangle if we rewrite the values of the sides with denominators of 3.  The numerators are in the relationship 12-16-20 which "reduces" to 3-4-5.  Watch for this fractional type or shortcut.