1. Right Triangles and Trigonometry
Geometry Chapter #9
Right Triangles and Trigonometry

2. Similar Right Triangles
Lesson #9.1
Similar Right Triangles

3. Right Triangles All right triangles have one right angle.
The longest side of a right triangle is the
hypotenuse. It is always opposite the right angle.
The sides that form the right angle are called legs.

4. Altitude of a Triangle Every triangle has 3 altitudes.
An altitude of a triangle is the perpendicular segment
from a vertex to the opposite side
(or to a line that contains the opposite side).
Every triangle has 3 altitudes.

5. The three altitudes of a triangle are concurrent.
The point of concurrency of the altitudes
is called the orthocenter of the triangle.
The orthocenter may lie inside, on, or outside the triangle.
In an acute triangle, the
orthocenter is
INSIDE the triangle.
In an obtuse triangle, the
orthocenter is
OUTSIDE the triangle.
In a right triangle, the
orthocenter is
ON the triangle.

6. Theorem #9.1
If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

7. Theorem #9.2
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.
The length of the altitude is the geometric mean of the lengths of the two segments.

8. Theorem #9.3
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments.
The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

9. The Pythagorean Theorem
Lesson #9.2
The Pythagorean Theorem

10. Theorem #9.4 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞 𝟐 = 𝐥𝐞𝐠 𝟐 + 𝐥𝐞𝐠 𝟐 𝐜 𝟐 = 𝐚 𝟐 + 𝐛 𝟐
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞 𝟐 = 𝐥𝐞𝐠 𝟐 𝐥𝐞𝐠 𝟐
𝐜 𝟐 = 𝐚 𝟐 𝐛 𝟐

11. 25 9 16 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞 𝟐 = 𝐥𝐞𝐠 𝟐 + 𝐥𝐞𝐠 𝟐 𝐜 𝟐 = 𝐚 𝟐 + 𝐛 𝟐 𝟓 𝟐 = 𝟑 𝟐 + 𝟒 𝟐
𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞 𝟐 = 𝐥𝐞𝐠 𝟐 𝐥𝐞𝐠 𝟐 25
𝐜 𝟐 = 𝐚 𝟐 𝐛 𝟐 9 3 5
𝟓 𝟐 = 𝟑 𝟐 𝟒 𝟐 4
25 = 16
25 = 25

12. Pythagorean Triples
Pythagorean Triples are sets of whole numbers,
a, b, and c, that work in the Pythagorean Theorem.
Here are some that you should memorize:
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
ANY multiples of these Triples are also Triples.
(ex: 6, 8, 10---multiple 3, 4, 5 each by 2)

13. The Converse of the Pythagorean Theorem
Lesson #9.3
The Converse of the Pythagorean Theorem

14. Theorem #9.5 𝐜 𝟐 = 𝐚 𝟐 + 𝐛 𝟐 𝐈𝐟 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞 𝟐 = 𝐥𝐞𝐠 𝟐 + 𝐥𝐞𝐠 𝟐 ,
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
𝐈𝐟 𝐡𝐲𝐩𝐨𝐭𝐞𝐧𝐮𝐬𝐞 𝟐 = 𝐥𝐞𝐠 𝟐 𝐥𝐞𝐠 𝟐 ,
then the triangle is a right triangle.
𝐜 𝟐 = 𝐚 𝟐 𝐛 𝟐
This theorem can be used to prove or
verify that a triangle is a right triangle.

15. Theorem #9.6
If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.
𝐈𝐟 𝐜 𝟐 < 𝐚 𝟐 𝐛 𝟐 ,
the triangle is acute.
𝟏𝟒 𝟐 ? 𝟗 𝟐 𝟏𝟏 𝟐
196 ? 𝟖𝟏 +121
The triangle
is acute.
196 < 𝟐𝟎𝟐

16. Theorem #9.7
If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.
𝐈𝐟 𝐜 𝟐 > 𝐚 𝟐 𝐛 𝟐 ,
the triangle is obtuse.
𝟐𝟏 𝟐 ? 𝟖 𝟐 𝟏𝟓 𝟐
441 ? 𝟔𝟒
The triangle
is obtuse.
441 > 𝟐𝟖𝟗

17. Special Right Triangles
Lesson #9.4
Special Right Triangles

18. Special Right Triangle Theorems
In a right triangle, the hypotenuse is 𝟐 times as long as each leg.
7 𝟐

19. Special Right Triangle Theorems
In a right triangle, the hypotenuse is twice as long as the shorter leg
and the longer leg is 𝟑 times as long as the shorter leg.

20. Lesson #9.5
Trigonometric Ratios

21. 6 Trigonometric Functions
For angle x in the right triangle above, the 6 trig ratios are
sin x = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 csc x = 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆
cos x = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 sec x = 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕
tan x = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 cot x = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆

22. A mnemonic device that you can use to memorize the
definition of the first three trig functions is SOH CAH TOA.

23. Another mnemonic device that you can use to memorize the
definition of the first three trig functions is

24. 𝜃 sin 𝜃 = csc 𝜃 = cos 𝜃 = sec 𝜃 = tan 𝜃 = cot 𝜃 = 𝟕 𝟐𝟓 𝟐𝟓 𝟕 𝟐𝟒 𝟐𝟓
𝟐𝟓 𝟐𝟒
𝟕 𝟐𝟒
𝟐𝟒 𝟕

25. Example: In Δ ABC, AB = 3, BC = 4, find AC.
Then evaluate the 6 trig functions of angle 𝑨.
First, how could you find the length of hypotenuse AC?
Use the Pythagorean Theorem:
𝑎 𝑏 2 = 𝑐 2
= 𝐴𝐶 2
= 𝐴𝐶 2
25= 𝐴𝐶 2
5= AC

26. Example: In Δ ABC, AB = 3, BC = 4, find AC.
Then evaluate the 6 trig functions of angle 𝑨.
hypotenuse
opposite
adjacent
Next, how can you evaluate the 6 trig functions of angle A?
Use the trig ratios from angle A’s point of view:
sin A = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 = 4 5
csc A = 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 = 5 4
cos A = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 = 3 5
sec A = 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 = 5 3
tan A = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 = 4 3
cot A = 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 = 3 4

27. Solving Right Triangles
Lesson #9.6
Solving Right Triangles

28. Solving a right triangle
Example: Solve the right triangle to find x, h, and missing angle A.
Since this is not a special right triangle, I must use my calculator to find the values of the trig functions of the angles.
IMPORTANT:
My calculator must be in degrees (not radians) .
Round trig values correctly to 4 decimal places, and
round final side lengths or angles to 1 decimal place. A
hyp
62° h
opp
sin 28° = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
tan 28° = 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 C
adj B
sin 28° = 12 ℎ
tan 28° = 12 𝑥 𝑥 ℎ ℎ 𝑥
Since two angles are
28 and 90, the third
angle (angle A) must
be 62 because the 3
angles must add up
to 180 and
= 62.
h sin 28° =
x tan 28° =
sin 28°
sin 28°
tan 28°
tan 28°
x =
h =
h = 25.6
x = 22.6

29. Solving a right triangle
Solving a right triangle means finding ALL missing side lengths
AND angle measures.
Example: Solve the right triangle to find x, y, and missing angle A.
Now notice that this is a right triangle.
I can use the special right triangles short-cut to solve
this problem.
30°
hyp
opp
sin C = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos C = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
adj
sin 60 = 𝑥 10
cos 60 = 𝑦 10
3 2 = 𝑥 10
1 2 = 𝑦 10
Since two angles are
60 and 90, the third
angle (angle A) must
be 30 because the 3
angles must add up
to 180 and
= 30.
2y = 10
2x = 10 3 2 2 2 2
x = 5 3
y = 5

30. Example in real life:
Wind speed affects the angle at which a kite flies. You are flying a kite 4 feet off the ground using 500 feet of line. At a wind speed of 35 miles per hour, the kite will make an angle with a line parallel to the ground of 48 degrees. How high off the ground is the kite.
sin 48 = 𝑑 500
(0.7431) = 𝑑 500
372 ≈ d
Add on the 4 feet for the height at which you are holding it, and the kite is about 376 feet off the ground.

31. Find the measure of angle 𝑪 for the triangle shown.
𝐶= 𝑐𝑜𝑠 −
= 𝑐𝑜𝑠 −
= 𝑐𝑜𝑠 −
≈36.9°