Geometry Module 1 Review Notes

Coordinate Geometry: Midpoint and Distance Formulas

The midpoint of a line segment is the point that divides the segment into two congruent segments. On a two-dimensional coordinate plane, the midpoint $M$ of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the formula: $(M_x, M_y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. If you are given the midpoint and one endpoint, the coordinates of the missing endpoint can be found by rearranging the formula: $x_2 = 2M_x - x_1$ and $y_2 = 2M_y - y_1$.

The distance formula is used to find the length of a line segment between two points $(x_1, y_1)$ and $(x_2, y_2)$. It is derived from the Pythagorean Theorem and is expressed as: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

In the case of segment $TS$ with endpoints $T(1, -4)$ and $S(-5, 16)$, the midpoint $M$ is calculated as $\left(\frac{1 + (-5)}{2}, \frac{-4 + 16}{2}\right) = (-2, 6)$. The length of the entire segment $TS$ is $\sqrt{(-5 - 1)^2 + (16 - (-4))^2} = \sqrt{(-6)^2 + (20)^2} = \sqrt{36 + 400} = \sqrt{436} \approx 20.9$. Consequently, the length of segment $MS$ is half of $TS$, which is approximately $10.4$.

For a segment $JH$ with endpoints $J(-11, 8)$ and $H(9, -13)$, the length is computed as: $JH = \sqrt{(9 - (-11))^2 + (-13 - 8)^2} = \sqrt{(20)^2 + (-21)^2} = \sqrt{400 + 441} = \sqrt{841} = 29$. The midpoint of $JH$ is $\left(\frac{-11 + 9}{2}, \frac{8 + (-13)}{2}\right) = (-1, -2.5)$.

Fundamental Geometric Definitions and Notations

Geometry relies on several undefined terms that serve as the building blocks for all other concepts. A point is a location in space with no size, often represented by a dot. A line is a straight path that extends infinitely in two opposite directions. A plane is a flat surface that extends infinitely in all directions. Collinear points are points that lie on the same line. Coplanar points are points that lie in the same plane.

In practical visual terms, a pixel or dot on a computer screen represents a point. The computer screen itself represents a plane. The edge of the computer screen represents a line segment. For a cube like $DEFGHIJK$, the intersection of two planes (such as plane $DEFC$ and plane $FJIE$) is always a line (in this case, line $FE$). The intersection of two lines (such as line segment $DE$ and line segment $EI$) is a single point (Point $E$).

Naming conventions in geometry are specific. A line can be named using any two points on the line, such as $\overleftrightarrow{AB}$, $\overleftrightarrow{BA}$, or by a single lowercase script letter. A line segment is denoted as $\overline{AL}$. These notations are not interchangeable: $\overline{AL}$ refers to the physical segment between points $A$ and $L$, $AL$ represents the numerical length of that segment, and $\overleftrightarrow{AL}$ represents the infinite line passing through those points.

The Segment Addition Postulate and Algebraic Algebra

The Segment Addition Postulate states that if point $B$ is between points $A$ and $C$, then the sum of the lengths of the smaller segments equals the length of the whole segment: $AB + BC = AC$. This principle allows for solving missing measures through algebraic equations.

For example, if point $Y$ is between points $X$ and $Z$, where $XY = 4a + 17$, $YZ = 22$, and $XZ = 10a - 3$, we set up the equation: $(4a + 17) + 22 = 10a - 3$. Simplifying this gives $4a + 39 = 10a - 3$, which yields $6a = 42$ or $a = 7$. Substituting this back, $XZ = 10(7) - 3 = 67$.

In another scenario, if segments are congruent, their lengths are equal. In segment $ABCD$, where $AB = 7x - 13$, $BC = 3x + 37$, and $CD = 57 - 3x$, and visual markings indicate $AB \cong CD$, we solve for $x$ by setting $7x - 13 = 57 - 3x$. This results in $10x = 70$, so $x = 7$. The segments would be $AB = 36$, $BC = 58$, and $CD = 36$. The total length $AD = 36 + 58 + 36 = 130$.

When a midpoint is involved, the two smaller segments are equal to each other and half of the whole. If $K$ is the midpoint of segment $JL$, then $JK = KL$. Given $JK = 3(x + 19)$ and $JL = 16x + 8$, the equation is $2[3(x + 19)] = 16x + 8$. This simplifies to $6x + 114 = 16x + 8$, leading to $10x = 106$ and $x = 10.6$. The length of $KL$ is then $3(10.6 + 19) = 3(29.6) = 88.8$.

Number Line Observations and Calculations

Finding midpoints and lengths on a 1D number line is simpler. The length is the absolute difference between coordinates, $|a - b|$. The midpoint is the average of the coordinates, $\frac{a + b}{2}$.

On a number line with points $J(-7)$, $K(-3)$, $L(-1)$, $M(1)$, $N(3)$, and $P(5)$, various midpoints can be calculated:

• Midpoint of $KM$: $\frac{-3 + 1}{2} = -1$
• Midpoint of $LN$: $\frac{-1 + 3}{2} = 1$
• Midpoint of $JN$: $\frac{-7 + 3}{2} = -2$

For another number line with points $E(-6)$, $F(-2)$, $G(0)$, $H(4)$, $J(6)$, $K(8)$, and $L(10)$, midpoints are as follows:

• Midpoint of $FK$: $\frac{-2 + 8}{2} = 3$
• Midpoint of $EF$: $\frac{-6 + (-2)}{2} = -4$
• Midpoint of $EL$: $\frac{-6 + 10}{2} = 2$

Real-World Applications of Midpoints

Midpoint concepts are frequently applied to spatial planning and distance estimation. For instance, if Callie wants to build a fence halfway between her house (located at the $10\,yd$ mark) and her neighbor's house (located at the $28\,yd$ mark), the distance of the fence from her home is calculated by finding the midpoint between the two locations: $\frac{10 + 28}{2} = 19\,yd$. The distance from Callie's house to the fence would be $19 - 10 = 9\,yd$.

In a dining scenario, Calvino's home is the midpoint between Fast Pizza and Pizza Now. If Fast Pizza is a quarter mile ($0.25\,mi$) away from Calvino's house, then Pizza Now is also a quarter mile ($0.25\,mi$) away in the opposite direction. The total distance between the two pizzerias is the sum of these distances, which is $0.25 + 0.25 = 0.5\,mi$.