Chapter 1: Triangles and Circles
Chapter 1 Summary and Review
Key Concepts
- The sum of the angles in a triangle is [latex]180°{.}[/latex]
- A right triangle has one angle of [latex]90°{.}[/latex]
- All of the angles of an equilateral triangle are equal.
- The base angles of an isosceles triangle are equal.
- Vertical angles are equal.
- If parallel lines are intersected by a transversal, the alternate interior angles are equal. Corresponding angles are also equal.
- Two triangles are congruent if they have exactly the same size and shape.
- The altitude of an equilateral triangle divides it into two congruent right triangles.
- In a [latex]30°-60°-90°[/latex] right triangle, the leg opposite the [latex]30°[/latex] angle is half the length of the hypotenuse.
- Two triangles are similar if they have the same shape but not necessarily the same size. The corresponding angles are equal, and the corresponding sides are proportional.
- Similar Triangles.
Two triangles are similar if either- their corresponding angles are equal, or
- their corresponding sides are proportional.
- If two right triangles have one pair of corresponding acute angles with the same measure, then the triangles are similar.
- Distance Formula.
The distance [latex]d[/latex] between two points [latex]P_{1}(x_1, y_1)[/latex] and [latex]P_{2}(x_2, y_2)[/latex] is
[latex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/latex] - Any number that can be written as a quotient of two integers [latex]\dfrac{a}{b},~~ b\not=0,~~{,}[/latex] is called a rational number. The decimal form of a rational number is either a terminating decimal or a repeating decimal.
- An irrational number is one that cannot be written as a quotient of two integers [latex]\dfrac{a}{b},~~ b\not=0,~~{.}[/latex] We cannot write down an exact decimal equivalent for an irrational number.
- A circle is the set of all points in a plane that lie at a given distance, called the radius, from a fixed point called the center.
- Circle.
The equation for a circle of radius [latex]r[/latex] centered at the origin is
[latex]x^2+y^2=r^2[/latex] - The circle [latex]x^2 + y^2 = 1{,}[/latex] which is centered at the origin and has radius [latex]1[/latex] unit, is called the unit circle.
- Circumference of a Circle.
The circumference of a circle of radius [latex]r[/latex] is given by
[latex]C=2\pi r[/latex] - Area of a Circle.
The area of a circle of radius [latex]r[/latex] is given by
[latex]A=\pi r^2[/latex]