Similar Triangles & Trigonometry
What makes triangles similar?
Same shape: corresponding angles equal, sides proportional.
What is slope?
Slope is the rate of change (rise over run, m = Δy/Δx)
Solve: 3x = 12
x = 4
What is a linear system?
Two or more linear equations solved together to find a common solution.
What shape is a quadratic graph?
A parabola
In similar triangles, if one has sides 3 cm and 6 cm, and the other has a matching side of 6 cm, what is the other side?
6 cm is double 3, so other side = 12 cm
Find slope between (2, 3) and (4, 7).
m = (7 - 3)/(4 - 2) = 4/2 = 2
Solve: 2x + 5 = 11
x = 3
Solve: y = 2x + 1 and y = -x + 4.
Set equal: 2x + 1 = -x + 4 ⇒ x = 1, y = 3
Find vertex of y = (x - 3)² + 1
Vertex is (3, 1)
What is SOHCAHTOA used for?
To remember how to calculate sine, cosine, and tangent in right triangles.
Write equation with slope 2 and y-intercept -1.
y = 2x - 1
Solve: 5x - 7 = 3x + 9
x = 8
Solve by substitution: y = x + 2, 2x + y = 9
Sub: 2x + (x + 2) = 9 ⇒ x = 3, y = 5
Factor x² + 5x + 6
(x + 2)(x + 3)
A triangle has angle A = 35°, and the adjacent side = 10 cm. Find the opposite side.
tan(35°) = opp/10 ⇒ opp ≈ 7.0 cm
Find equation of line through (1, 2) and (3, 8).
m = 3 ⇒ y - 2 = 3(x - 1) ⇒ y = 3x - 1
A student scored 5 more than 3 times their friend's score. If the total is 29, find both scores.
Let x = friend's score: 3x + 5 = 29 ⇒ x = 8, 3x + 5 = 29 ⇒ student = 24
Solve by elimination: 3x + 2y = 16, x - 2y = 4
Add equations: 4x = 20 ⇒ x = 5, y = 0.5
Solve x² - 9 = 0
x = +3
A 1.5 m tall flagpole casts a shadow of 2.4 m. A nearby tree casts a 12 m shadow. How tall is the tree?
Use proportions: 1.5/2.4 = x/12 ⇒ x = 7.5 m
Graph a job paying $15/hour, no base pay.
y = 15x (x = hours, y = earnings)
Solve and check: 4(x - 1) = 2(x + 5)
4x - 4 = 2x + 10 ⇒ x = 7
Child tickets $6, adult $10. 2C + A = 22, C + A = 14. Find C and A.
Solve: C = 8, A = 6
h = -5t² + 20t. Max height?
Vertex: t = -b/2a = 2 ⇒ h = -5(2)² + 20(2) = 20 m