Triangles

Description:

Triangle questions require you to know and apply several triangle equations.

Approach:

While nearly all triangle information can be found at the beginning of every math section, a strong knowledge of the Pythagorean Theorem, special triangles, and basic triangle relationships is required to master these questions.

1) Area of a Triangle

The area of a triangle can be found by using the equation below where b represents the base of the triangle and h represents the height. Remember that the height must be perpendicular to the base.

1-1. Practice:

What is the area of a triangle with a height of 4 and a base of 8?
What is the height of a triangle with an area of 20 and a base of 4?

2) Perimeter of a Triangle

The Perimeter of a triangle is equal the sum of all of its sides.

3) Isosceles Triangles

Isosceles triangles have two equal sides and two equal angles.

4) Equilateral Triangles

Equilateral triangles have three equal angles and three equal sides.

5) Special Right Triangles

Special right triangles include the 30-60-90, the 45-45-90, the 3-4-5 and the 5-12-13.

30-60-90 triangles occur when an equilateral triangle is split in half. The resulting sides are always in a ratio of 1:√3:2

45-45-90 triangles occur when a square is split in half. The resulting sides are always in a ratio of 1:1:√2

5-1. Practice:

What is the height of a 30-60-90 triangle if its shortest side is length 6?
What is the length of a square's diagonal if its area is 100?

6) Pythagorean Theorem

The Pythagorean Theorem is used to determine a side of a right triangle if the other two sides are known. In the equation below, c is hypotenuse and a and b are the other two sides. The Pythagorean Theorem can only be used for right-triangles.

a² + b² = c²

6-1. Practice:

What is the hypotenuse of a right triangle with sides length 12 and 16?
What is shortest side of a right triangle whose hypotenuse is length 13 and whose other side length is 12?

7) Angle Side Relationship

Sides of a triangle are proportional to the angles that point to them. Thus, the largest angle in a triangle points to the longest side and the smallest angle points to the shortest side.

8) Size Limitations for a Triangle

Every side of a triangle must be longer than the difference between the other two sides, and shorter than the sum of the other two sides.
Thus, if two sides of a triangle are 5 and 8 units long respectively, the third side must be greater than 3 and less than 13.

8-1. Practice:

What is the largest possible integer value for the third side of triangle with sides length 6 and 8?
What is the shortest possible integer value for the third side of a triangle with sides length 3 and 7?

9) Similar Triangles

Similar triangles are triangles with equal angles. If two triangles have three common angles, then the ratio of any side of one triangle to the corresponding side of the other triangle is equal. Suppose that triangle ABC is similar to triangle DEF in such a way that the angle at vertex A is equal to the angle at vertex D, the angle at B is equal to the angle at E, and the angle at C is equal to the angle at F. Then, once this is known, it is possible to deduce proportionalities between corresponding sides of the two triangles, such as the following:

^AB⁄_DE = ^BC⁄_EF = ^CA⁄_FD

9-1. Practice:

Triangles ABC and DEF are similar. If AB is 5, DE is 10, and BC is 6, what is the length of EF?
Triangles ABC and DEF are similar. If the perimeter of ABC is twice the length of the perimeter of DEF, what is the length of the shortest side of triangle ABC if the shortest side of triangle DEF is 10?

Triangles

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