M151Q Precalculus · Unit 3: Quadratics, polynomials, triangles · on Exam 3
TRIG 1.1–1.3 Angles, triangles, similar triangles, and circles
Study notes for Angles, triangles, similar triangles, and circles in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- Angles in a triangle add to $180^\circ$. Right triangles obey the Pythagorean theorem $a^2 + b^2 = c^2$. Isosceles triangles have equal base angles.
- Similar triangles have equal angles and proportional sides: corresponding sides are in the same ratio. Set up proportions with matching positions (short/short = long/long). Common sources: parallel lines, shadows, nested right triangles.
- Special right triangles: $45^\circ$–$45^\circ$–$90^\circ$ sides $1 : 1 : \sqrt2$; $30^\circ$–$60^\circ$–$90^\circ$ sides $1 : \sqrt3 : 2$.
- Circles: circumference $2\pi r$, area $\pi r^2$; the equation of a circle centered at $(h,k)$ is $(x-h)^2 + (y-k)^2 = r^2$. Distance formula is the Pythagorean theorem in coordinates.
Key formulas
Triangles
$$A + B + C = 180^\circ,\qquad a^2 + b^2 = c^2$$
Similar triangles
$$\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$$
Circles
$$(x-h)^2 + (y-k)^2 = r^2,\qquad d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
Worked example
A 1.8 m person casts a 2.4 m shadow while a tree casts a 14 m shadow. How tall is the tree?
Show the solution
Similar triangles: $\dfrac{h}{14} = \dfrac{1.8}{2.4} \Rightarrow h = 10.5$ m.
Common mistakes
- Matching non-corresponding sides in a proportion.
- Using the Pythagorean theorem on a triangle with no right angle.
Exam tipRedraw the two similar triangles separately and in the same orientation before writing the proportion.
Keep going in StudyCat
StudyCat is free for Montana State students: an endless quizzer with worked explanations, one-question lessons, flashcards, exam prep checklists, the class calendar with every deadline, a grade calculator and a study board for M151Q.