Unit 7 Part 1:Right Triangles and Trigonometry

By:Danielle Gardon and Sandeep Menon

Key Concepts● Simplifying radical expressions● Determining whether a triangle is obtuse, acute or right using

the Pythagorean Theorem and given the length of three sides or the coordinates of the vertices

● Determine the length of a side of a right triangle using the Pythagorean Theorem and solving multistep problems using the Pythagorean Theorem (for example, finding the perimeter or area of a triangle)● Calculating the missing sides of 45-45-90 or 30-60-90 triangles given the length of at least one of its sides.

● Using the relationships among the sides of special right triangles in multistep problems, such as finding the height of a trapezoid, perimeter or a parallelogram, or the area of a triangle.

Common Mistakes

● Forgetting the Pythagorean triples● Forgetting the special right triangle

rules● Forgetting how to use the Pythagorean theorem to classify triangles● Applying the Pythagorean theorem in 3-D, like in a pyramid

Connections● In Unit 7 Part 2, the Pythagorean theorem can be

used with SOHCAHTOA to solve a triangle● In Unit 10, right triangles need to be used to find

side lengths and apothems before finding the area● In Unit 11, sometimes you need to find a height or

a length or a radius with a special right triangle or the triples before finding volume and/or surface area

Identify the unknown side as a leg or hypotenuse. Then, find X. Write your answer in simplest form.

Example 1: Using the Pythagorean Theorem

You know the answer is hypotenuse because the missing side isn’t opposite of the right angle. You know the answer is 12 because in this case the triangle is a Pythagorean triple (5-12-13).

132-52=x2

169-25=x2

√144=x2

x=12

5

The answers are: leg and 12.

13

X

Identify the missing sides of the triangle.

18√3 y

x

Example 2: Special Right triangles

60°

We know that since there are 180° in a triangle, the unknown angle is 30°. Therefore, this triangle is a 30-60-90 triangle. This means that we know that y, being opposite of the 30° angle is ½ of 18√3, since the hypotenuse in a 30-60-90° triangle is twice that of the side opposite of 30°. We can figure out that x=27 because the side opposite the 60° is always √3 times the side opposite the 30° angle (y) in a 30-60-90 triangle.

The answers are: x=27 and y=9√3

Given the diagram, solve for x, y, and z.

x y

z 15√2

Example 3: Using similar right triangles

5√6

You can solve for y by using a 30-60-90 triangle. You know that the 5√6 side, the 15√2 side, and the y side create a 30-60-90 triangle because you know that y is the hypotenuse through the fact that the angle opposite of y is 90°, since straight lines form straight angles, you subtract the given right angle next to the angle (90°) from 180° to get 90°. You know that 180-(90+30) gives the the remaining angle of the triangle, 60°. This allows you to conclude that y is 2 times 5√6 and that y=10√6. You can solve for x and z by the fact that you can use a 30-60-90 triangle, since the other 2 angles of the largest triangle in the figure are 90 and 30, you can conclude that the remaining angle is 60°,which allows you to conclude in the smallest triangle formed that the last angle must be 30°. The smallest triangle is proportional to the medium triangle through AAA similarity. This way, you can conclude through the 2 given sides that the larger triangle is √3 times the smaller triangle, allowing you to conclude that z is 5√6 divided by √3(which is 5√2), and x is twice z, so x=10√2.

The answers are: x=10√2, y=10√6,z=5√2

30°

A telephone pole perpendicular to the ground is 20 feet high. A wire running from the top of the pole to the ground is 52 feet long. How far is the end of the wire from the base of the pole?

Real Life Situation

20 feet

52 feet

X

Answer:48 feetYou can solve this by using Pythagorean triples to find that this is a 5-12-13 triangle on a x4 base, making the missing side 48. You can also use the Pythagorean theorem like this:202+x2=522

400+x2=2704x2=2304x=48