Pythagorean
Theorem
Theorem
Find the missing side: A right triangle has legs 4.5 and 3. What is the hypotenuse rounded to the nearest tenth?
(4.5)2 + 32 = 20.25 + 9 = 29.25
(square root 29.25) is about 5.4
The hypotenuse is 5.4.
In a 45–45–90 triangle, if one leg is 7, what is the hypotenuse?
7 * (square root 2) = 7(square root 2)
State the geometric mean of 4 and 9.
(square root 4 * 9) = (square root 36) = 6
Find sin(30 degrees).
1/2 or 0.5
Define “hypotenuse.”
The side opposite the right angle and the longest side.
A triangle has sides 5, 9.4, and (square root 113.36). Use the converse of the Pythagorean Theorem to determine if it is a right triangle.
52 + (9.4)2 = 25 + 88.36 = 113.36
(square root 113.36)2 = 113.36
113.36 = 113.36, right triangle
In a 30–60–90 triangle, the leg across from the 30 degree angle is 9. What is the length of the hypotenuse?
9 * 2 = 18
In a right triangle, the altitude to the hypotenuse splits it into segments of 25 and 9. Find the altitude.
In a right triangle, tan(θ) = 3/4. If the adjacent side is 8, find the opposite side.
tan is opposite/adjacent, so 3/4 relates to x/8. Multiplying 3/4 by 2/2 you get 6/8. Thus, x = 6 so the opposite side is 6.
Write each of the three trig functions as a fraction to how they correspond to their position relative to the triangle. For example, one is adjacent/hypotenuse.
sin: opposite/hypotenuse
cos: adjacent/hypotenuse
tan: opposite/adjacent
A triangle has sides 10, 15, and 18. Determine whether it is acute, right, or obtuse using the relationship among the squares of the sides.
102 + 152 = 100 + 225 = 325
182 = 324
325 > 324, acute triangle
In a 45–45–90 triangle, the hypotenuse is 10(square root 2). What is the length of each leg?
Suppose a triangle is split by its altitude and forms two similar triangles. On the smaller triangle that was just made, if the hypotenuse is 6, find the leg that is not the altitude. The original triangle's hypotenuse is 10.
62 = x * 10
36 = 10x
3.6 = x
The lg that is not the altitude is 3.6.
Find A if cos(A) = 3/5 to the nearest degree.
cos-1(3/5) is around 53 degrees
Explain the difference between an acute triangle and an obtuse triangle using side length relationships.
Acute: a2 + b2 > c2
Obtuse: a2 + b2 < c2
A ladder leans against a wall so the bottom is 9 ft from the wall, and the ladder is 15 ft long. How high up the wall does the ladder reach?
92 + x2 = 152
81 + x2 = 225
x2 = 144
x = 12
The ladder reaches 12 feet up the wall.
A 30–60–90 triangle has hypotenuse 26. Find the longer leg. What is the degree this leg is across from?
Since the shorter side is half the hypotenuse, it would be 13. The longer leg is across from a 60 degree angle and is found by multiplying the shorter leg by (square root 3). Thus, the longer leg is 13(square root 3).
Suppose a hypotenuse is split by an altitude. One part is 12 and the other is x. If the altitude is 18, find x.
18 = (square root 12 * x)
182 = 12x
324 = 12x
27 = x
If sin(B) = 3/5 and cos(B) = 4/5, what is tan(B)?
tan(B) = sin(B)/cos(B)
tan(B) = (3/5)/(4/5)
tan(B) = 3/4
Define “geometric mean” and give an example of when it is used in right triangle similarity.
A number x where x = (square root a * b). It is used to find altitudes or legs formed when a right triangle’s hypotenuse is split by its altitude.
A triangle has side lengths x, 20, and 21, with the angle opposite side of x being a right angle. Find x. Then state whether the triangle would still be right if the side opposite the right angle were increased by 1.
202 + 212 = x2
400 + 441 = x2
841 = x2
29 = x
The side opposite of the right angle is the hypotenuse, and if it were increased by 1 c2 would increase, making it change to an obtuse triangle.
A triangle has sides 15, 15, and 15(square root 2). Verify that this is a special right triangle and determine all angle measures.
This is a 45–45–90 triangle where there are two angle measures of 45 degrees, and the 90 degree right angle.
In a right triangle with lengths 3, 4, and 5, what is the altitude?
h/4 = 3/5
h = 12/5
Thus, the altitude is 12/5 or 2.4.
A right triangle has a leg opposite to angle C equal to 14 and the hypotenuse equal to 20. Find angle C to the nearest degree, then find the adjacent leg to the nearest tenth.
C = sin-1(14/20) which is around 44 degrees.
142 + x2 = 202
196 + x2 = 400
x2 = 204
x is around 14.3.
In words, what is the altitude of a right triangle and how do you find it?
The altitude is the line from the right angle to the hypotenuse that splits the hypotenuse. You find it by calculating the geometric mean of the two values the hypotenuse was split into, or by setting up a ratio with the legs and hypotenuse.