Triangles

Angles in a Triangle

 

A triangle is a plane figure with three sides.  All of the angles in a triangle add up to 180°.  Angles also have a relationship with the side across from them:  The larger the angle, the larger the side across from it will be in relation to the other sides.  If two angles are the same, the sides across from them will be the same length.  Also, if an angle is 90° or greater, it must be the largest angle in the triangle and the side across from it must be the longest side.

 

Area and Perimeter of Triangles

 

The formula for area of a triangle is A = bh, or half of the base times the height.  The height of the triangle is not the length of any side of the triangle, but the distance the furthest point it from the side you are using as the base. 

 

Classifying Triangles by their Sides

Triangles can be classified two ways:  by their angles or by their sides.  If you have all the information about a triangle, you should be able to give it two classifications.  Classifying by sides, there are three types of triangles:  equilateral, isosceles, and scalene.  An equilateral triangle has three equal sides.  Since all three sides are equal, all three angles are equal:  60°.  An isosceles triangle has two equal sides, which means it also has two equal angles.  A scalene triangle has no equal sides or angles.

 

Classifying Triangles by their Angles

 

Triangles classified by angles also have three types:  acute, right and obtuse.  An acute triangle has all of its angles under 90°.  An equilateral triangle is always acute.  A right triangle has a right angle, which is the largest angle in the triangle.  The long side across from the right angle is called the hypotenuse and the other two sides are called the legs.  An obtuse triangle has one angle that is greater than 90°.

 

Pythagorean Theorem

 

In a right triangle, there is a way to find the length of a missing side if we have the length of the other two sides.  The Pythagorean Theorem is a formula you have on your formula sheet, and it says a2 + b2 = c2, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse.  If you are given a right triangle like this:


 


What is the length of the missing side of this right triangle?  Fill in the values you know in the formula:  a2 + b2 = c2.

 

32 + b2 = 52

Simplify the exponents:

9 + b2 = 25

Solve for b:

9 + b2 = 25

-9        -9

b2 = 16

(Square root each side)

b = 4

 

This is a classic 3-4-5 right triangle.  Test creators like it because it works out into three perfect whole numbers.  You may see other variations on this, like 6-8-10.