Pythagorean Geometry – Tutorial and Practice Questions

A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1).  A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

Questions on basic geometry and Pythagorean geometry are on most high school tests

If we have a right triangle ABC, where its sides (legs) are a and b and c is a hypotenuse (the side opposite the right angle), then we can establish a relationship between these sides using the following formula:

This formula is proven in the Pythagorean Theorem. There are many proofs of this theorem, but we’ll look at just one geometrical proof:

If we draw squares on the right triangle’s sides, then the area of the square upon the hypotenuse is equal to the sum of the areas of the squares that are upon other two sides of the triangle. Since the areas of these squares are a², b² and c², that is how we got the formula above.

One of the famous right triangles is one with sides 3, 4 and 5. And we can see here that:

3² + 4² = 5²

9 + 16 = 25

25 = 25

Example:
We write the information we have about triangle ABC and we draw a picture of it for better understanding of the relation between

its elements:

P = 18 cm

a – b = 3 cm

h=?

We use the formula for the perimeter of the isosceles triangle, since that is what is given to us:

P = a + 2b = 18 cm

Notice that we have 2 equations with 2 variables, so we can solve it as a system of equations:

a + 2b = 18

a –  b = 3               / multiply by 2

a + 2b = 18

2a – 2b = 6   

a + 2b + 2a – 2b = 18 + 6

3a = 24

a = 24:3 = 8 cm

Now we go back to find b:

a – b = 3

8 – b = 3

b = 8 – 3

b = 5cm

 The isosceles triangle ABC has a perimeter of 18 centimeters, and the difference between its base and legs is 3 centimeters. Find the height of this triangle.

We write the information we have about triangle ABC and we draw a picture of it for better understanding of the relation between

its elements:

P = 18 cm

a – b = 3 cm

h = ?

We use the formula for the perimeter of the

isosceles triangle, since that is what is given to us:

P = a + 2b = 18 cm

Notice that we have 2 equations with 2 variables, so we can solve it as a system of equations:

a + 2b = 18

a –  b = 3               / multiply by 2

a + 2b = 18

2a – 2b = 6   

a + 2b + 2a – 2b = 18 + 6

3a = 24

a = 24:3 = 8 cm

Now we go back to find b:

a – b = 3

8 – b = 3

b = 8 – 3

b = 5cm

Using Pythagorean Theorem, we can find the height using a and b, because the height falls on the side a at the right angle. Notice that height cuts side a exactly in half, and that’s why we use in the formula a/2. In this case, b is our hypotenuse, so we have:

Common geometry questions on on standardized tests :

• Solve for the missing angle or side
• Finding the area or perimeter of different shapes (e.g. triangles, rectangles, circles)
• Problems using the Pythagorean Theorem
• Calculate properties of geometric shapes such as angles, right angles or parallel sides
• Calculating volume or surface area of complex shapes for example spheres, cylinders or cones
• Solve geometric transformations such as rotation, translation or reflections

1. Not clearly labeling or identifying the given and unknown information in a problem
2. Not understanding the properties and definitions of basic geometric figures (e.g. line, angle, triangle, etc.)
3. Incorrectly using basic formulas (e.g. area of a triangle, Pythagorean theorem)
4. Incorrectly interpreting geometric diagrams
5. Not understanding the relationship between parallel lines and transversals
6. Not understanding the relationship between angles and their degree measures
7. Not understanding the relationship between perimeter and area