The Pythagorean Theorem is a relation in Geometry between the 3 sides of a right triangle The theorem states: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs NEXT
Generally speaking, the formula is written as: a 2 + b 2 = c 2 where a and b are the legs, and c is the hypotenuse. a c b See examples Practice a problem BACK
24 7 x 7 2 + 24 2 = x 2 49 + 576 = x 2 625 = x 2 25 = x 5 2 + x 2 = 13 2 25 + x 2 = 169 x 2 = 144 x = 12 x 13 5 Practice a Problem
A special right triangle is a right triangle whose sides are in a particular ratio. Recognizing special right triangles in geometry can help you to answer some questions quicker. Types of special triangles
There are 2 main types of special right triangles: 1) The 45-45-902) The 30-60-90 45 90 60 30 Click on the triangle you want to learn about.
The lengths of the sides of a 45°- 45°- 90° triangle are in the ratio of 1 : 1 : 45 90 √2 x x x Try a practice problem! Check out the 30-60-90!
The lengths of the sides of a 30°- 60°- 90° triangle are in the ratio of 1 : : 2 90 60 30 √3 x x√3 2x Try a practice problem! Check out the 45-45-90!
Solve for the missing parts of the triangle. Round your answers to the nearest tenth. 45 90 x y 7.4 z Click here to check your answers! Back to help
9030 x y 13 z Solve for the missing parts of the triangle. Round your answers to the nearest tenth. Click here to check your answers! Back to help
x = 7.4 y = 10.5 z = 45 Try it again! Try a 30-60-90 problem! Watch a helpful video! Or click here to go home and try a new section!
x = 6.5 y = 11.3 z = 60 Watch a helpful video! Try it again! Try a 45-45-90 problem! Or click here to go home and try a new section!
Right triangles have 3 special formulas that ONLY WORK in right triangles. They are: sin = cos = tan = Opposite hypotenuse adjacent hypotenuse opposite adjacent A BC a b c How to setup a trig problem See an example Go straight to try a problem!
Draw a picture depicting the situation. Be sure to place the degrees INSIDE the triangle. Place a stick figure at the angle as a point of reference. Thinking of yourself as the stick figure, label the opposite side (the side across from you), the hypotenuse (across from the right angle), and the adjacent side (the leftover side). Figure out which pair of sides the problem deals with (for example: opposite and hypotenuse) and choose the correct equation (in our example, sin) See an exampleTry a problem!
In right triangle ABC, hypotenuse AB=15 and angle A=35º. Find leg BC to the nearest tenth. 1) First, draw a picture, and label everything you know. 2) Then, figure out which trig function we will use. In this case, we will use sin. 3) Set up the equation. 4) Solve for x. sin 35 = X 15 x = 15 sin 35 x = 8.6 See how to set up a problem Try a problem!
In triangle RST, angle R is a right angle, angle S has measure of 65 degrees, and RS = 9. Find the measure of ST. A) 3.8B).05 C) 27.1D) 21.3
You have finished this part of the review. Now, either go back, and choose a new topic, or click here for more practice with this topic!here
cos 65 = x * cos 65 = 9 x = x = 21. 3 90 65 9 R S T 9x9x x 9 cos 65 Click here to go home and try something new! Try again?Back to explanation.
We use the LAW OF SINES to solve triangles that are not right triangle. The law of sines states the following: The sides of a triangle are to one another in the same ratio as the sines of their opposite angles. What does that mean? See a diagram explanation!
We can use this triangle to set up the equation….. A B C a b c sin A sin B sin C a b c == See how a problem is done. Try one on your own!
The three angles of a triangle are 40°, 75°, and 65°. When the side opposite the 75° angle is 10 cm, how long is the side opposite the 40° angle? Click to draw the triangle. Click again to set up the problem. Click a third time to see the answer! sin 75sin 40 10 x = x = 6.7 Now I’m ready to try one!
The three angles of a triangle are A = 30°, B = 70°, and C = 80°. If side a = 5 cm, find sides b and c. Click me to check your answer!
B = 9.4 C = 9.9 Did you get it right? YesNo Take me home so I can try something new! Take me back to the explanation again. Or click here for more Law of Sines practice!
Angles of elevation and depression are angles that are formed with the horizontal. If the line of sight is upward from the horizontal, the angle is an angle of elevation; if the line of sight is downward from the horizontal, the angle is an angle of depression. Using these types of angles and some trig, you can indirectly calculate heights of objects or distances between points.trig Alternatively, if the heights or distances are known, the angles can be determined. OK, I need to see how one’s done. Actually, I’m ready to try one on my own!
Suppose a flagpole casts a shadow of 20 feet. The angle of elevation from the end of the shadow to the top of the flagpole measures 50°. Find the height of the flagpole. Click once to draw the picture. Click again to setup the problem. Click a third time to see the answer! tan 50 = x 20 x = 23.8 NOW, I’m ready to try one on my own!
Suppose a tree 50 feet in height casts a shadow of length 60 feet. What is the angle of elevation from the end of the shadow to the top of the tree with respect to the ground? See the solution! Go back to the example…
First, we draw the picture Then, set up the equation: tan x = Finally, solve for x. x = tan -1 ( ) = 39.8 50 60 50 60 I need more practice! Click here to go home and try something new! Let me try again.
The Law of Cosines (also known as the Cosine Rule or Cosine Law) is a generalization of the Pythagorean TheoremPythagorean Theorem Basically, the Pythagorean Theorem requires there to be a right angle in a triangle But, if there is not, the Law of Cosines can be used Show me the Law of Cosines
For a triangle with sides a, b, and c opposite (respectively) the angles A, B, and C, the Law of Cosines states: c 2 = a 2 + b 2 - 2ab·cos(C) a 2 = b 2 + c 2 – 2bc·cos(A) b 2 = a 2 + c 2 - 2ac·cos(B) I need to see a picture I need to see an example Take me straight to a practice problem!
The law of cosines should be used when use have SSS or SAS in a triangle, like the one picture here… The law of cosines states: s 2 = r 2 + t 2 – 2rtcos(S), or r 2 = s 2 + t 2 - 2stcos(R), or t 2 = r 2 + s 2 – 2rscos(T) I still need to see an example. Great! Let me try one.
In triangle ABC, you are given a = 10, B = 32 o and c = 15. Find the measure of side b. First, write out the equation: b 2 = a 2 + c 2 -2 ac cos B. So, b 2 = 100 + 225 – 2*(10)*(15)*cos 32 º b 2 = 325 - 300 (0.848048096) b 2 = 325 - 254.4 b 2 = 70.59. Therefore, b = 8.4. Great! Let me try one. Take me back to the explanation.
In triangle ABC, you are given A = 28 o, b = 14, c = 10. Solve for side a. I can’t do it! Take me back to the example. I think I got it! Let me see the answer.
First, write out the equation: a 2 = b 2 + c 2 - 2bc*cos A. So, a 2 = 296 + 100 – 2*(14)*(10)*cos 28 º a 2 = 396 - 280 (0.8829475929) a 2 = 396 – 247.2 a 2 = 148.8. Therefore, a = 12.2. I got it! Take me home so I can try something different! I want to practice more! I didn’t get it. I want to try again.
You are now ready to take the chapter 8 test. Good Luck! End Show