Geometry Formulas and Properties review

A comprehensive set of practice questions covering coordinate geometry, triangle rules, trigonometry, transformations, circle theorems, and quadrilateral properties based on the lecture notes.

What is the distance formula in coordinate geometry?

$$d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

What is the slope formula ($m$)?

$$m = \frac{y_2-y_1}{x_2-x_1}$$

What is the relationship between the slopes of parallel lines?

They have the same slope.

How are the slopes of perpendicular lines related?

They are opposite reciprocal slopes.

What is the midpoint formula?

$$(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$$

What are the two forms for the equation of a line provided?

Slope-intercept form: $y = mx + b$ and Point-Slope Form: $y-y_1 = m(x-x_1)$

What is the formula for partitioning a line segment by factor $F$?

$$(x_1 + F(x_2-x_1), y_1 + F(y_2-y_1))$$

What are the coordinates of the Centroid for triangle $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$?

$$\text{Centroid} = (\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3})$$

What is the formula for the sum of the interior angles of a polygon with $n$ sides?

$$180(n-2)$$

What is the formula for one interior angle in a regular polygon?

$$\frac{180(n-2)}{n}$$

What is the sum of the exterior angles of any polygon?

$$360^\circ$$

What is the formula for one exterior angle in a regular polygon?

$$\frac{360}{n}$$

What is the altitude rule (PAAP) in right triangles?

$\frac{\text{partition 1}}{\text{altitude}} = \frac{\text{altitude}}{\text{partition 2}}$ ($\frac{AD}{CD} = \frac{CD}{DB}$)

What is the leg rule (HLLP) in right triangles?

$\frac{\text{whole hypotenuse}}{\text{leg}} = \frac{\text{leg}}{\text{projection}}$. This can be expressed as $\frac{AB}{AC} = \frac{AC}{AD}$ or $\frac{AB}{CB} = \frac{CB}{DB}$.

What does the External Angle Theorem state for triangles?

The sum of the non-adjacent interior angles is equal to the exterior angle.

What is the area formula for a triangle using trigonometry?

$$\text{Area} = \frac{1}{2}(a)(b)(\sin(C))$$

Under a reflection over the x-axis ($T_{x\text{-axis}}$), what does $(x,y)$ become?

$$(x, -y)$$

Under a rotation of $180^\circ$ ($R_{180}$), what does $(x,y)$ become?

$$(-x, -y)$$

What is the coordinate rule for a dilation $D_k$?

$$D_k(x, y) \rightarrow (kx, ky)$$

What is the formula for arc length ($s$)?

$$s = \frac{\theta}{360} \pi d$$

What is the formula for the area of a sector?

$$A_{\text{sector}} = \frac{\theta}{360} \pi r^2$$

What is the standard equation of a circle with center $(h, k)$ and radius $r$?

$$(x-h)^2 + (y-k)^2 = r^2$$

How do you calculate an angle formed inside a circle?

$$\text{inside } \angle = \frac{\text{Arc} + \text{Arc}}{2}$$

How do you calculate an angle formed outside a circle?

$$\text{outside } \angle = \frac{\text{Big} - \text{Small}}{2}$$

What is the segment relationship for two chords intersecting inside a circle?

$\text{Part} \times \text{Part} = \text{Part} \times \text{Part}$ ($a \cdot b = c \cdot d$)

What is the segment relationship for secants and tangents from an external point?

$$\text{Whole} \times \text{External} = \text{Whole} \times \text{External}$$

What is the formula for the median (midsegment) of a trapezoid?

$$b = \frac{a+c}{2}$$

List the five properties of a parallelogram.

1. Opposite sides are parallel. 2. Opposite sides are congruent. 3. Opposite angles are congruent. 4. Consecutive angles are supplementary. 5. Diagonals bisect each other.

What are the additional properties of a rhombus beyond those of a parallelogram?

1. Four congruent sides. 2. Diagonals bisect angles. 3. Diagonals are perpendicular.

What are the additional properties of a rectangle beyond those of a parallelogram?

1. Four right angles. 2. Diagonals are congruent.

What properties does a square possess?

A square possesses all properties of a parallelogram, rectangle, and rhombus.

What are the properties of an isosceles trapezoid?

All properties of a trapezoid plus: 1. Legs are congruent. 2. Base angles are congruent. 3. Opposite angles are supplementary. 4. Diagonals are congruent.

What are the five valid methods to prove triangles congruent?

SSS, SAS, ASA, AAS, and HL.

What are the three methods used to prove triangles similar ($\sim$)?

AA$, SAS$, and SSS$$

Which two combinations cannot be used to prove triangle congruence?

AAA and ASS.