Digital SAT Maths -Unit 4 - 4.8 Coordinate Geometry- Study Notes- New Syllabus
Digital SAT Maths -Unit 4 – 4.8 Coordinate Geometry- Study Notes- New syllabus
Digital SAT Maths -Unit 4 – 4.8 Coordinate Geometry- Study Notes – per latest Syllabus.
Key Concepts:
Distance formula
Midpoint formula
Equation of line
Distance Formula
The distance formula is used to find the distance between two points on the coordinate plane.
If the points are:
\( (x_1, y_1) \) and \( (x_2, y_2) \)
then the distance between them is:
\( d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2} \)
Where the Formula Comes From
The formula comes from the Pythagorean Theorem. The horizontal change \( (x_2-x_1) \) and vertical change \( (y_2-y_1) \) form a right triangle.
DIGITAL SAT Tip
The SAT frequently hides distance formula questions inside geometry problems such as finding:
- side length of a triangle
- radius of a circle
- whether a triangle is right
Example 1:
Find the distance between \( (1,2) \) and \( (5,6) \).
▶️ Answer/Explanation
\( d=\sqrt{(5-1)^2+(6-2)^2} \)
\( d=\sqrt{4^2+4^2} \)
\( d=\sqrt{16+16}=\sqrt{32}=4\sqrt{2} \)
Answer: \( 4\sqrt{2} \)
Example 2 (Triangle Side):
Find the length of the segment joining \( (-2,3) \) and \( (4,3) \).
▶️ Answer/Explanation
\( d=\sqrt{(4+2)^2+(3-3)^2} \)
\( d=\sqrt{6^2+0^2}=\sqrt{36}=6 \)
Answer: 6
Example 3 (Circle Radius SAT Style):
The center of a circle is \( (2,-1) \) and a point on the circle is \( (5,3) \). Find the radius.
▶️ Answer/Explanation
\( r=\sqrt{(5-2)^2+(3+1)^2} \)
\( r=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5 \)
Answer: radius = 5
Midpoint Formula
The midpoint of a line segment is the point exactly halfway between two endpoints.
If the endpoints are:
\( (x_1, y_1) \) and \( (x_2, y_2) \)
then the midpoint is:
\( M\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right) \)
Meaning
You are averaging the x-coordinates and averaging the y-coordinates.
DIGITAL SAT Tip
The SAT often hides midpoint questions inside geometry problems such as:
- center of a circle
- bisecting a segment
- finding missing coordinates
Example 1:
Find the midpoint of the segment joining \( (2,4) \) and \( (8,10) \).
▶️ Answer/Explanation
\( M=\left(\dfrac{2+8}{2},\dfrac{4+10}{2}\right) \)
\( M=(5,7) \)
Answer: \( (5,7) \)
Example 2 (Find Missing Endpoint):
The midpoint of a segment is \( (3,5) \) and one endpoint is \( (1,2) \). Find the other endpoint.
▶️ Answer/Explanation
Let the unknown point be \( (x,y) \).
\( \dfrac{1+x}{2}=3 \Rightarrow 1+x=6 \Rightarrow x=5 \)
\( \dfrac{2+y}{2}=5 \Rightarrow 2+y=10 \Rightarrow y=8 \)
Answer: \( (5,8) \)
Example 3 (Center of a Circle SAT Style):
The endpoints of a diameter of a circle are \( (-2,6) \) and \( (4,2) \). Find the center of the circle.
▶️ Answer/Explanation
The center of a circle is the midpoint of the diameter.
\( M=\left(\dfrac{-2+4}{2},\dfrac{6+2}{2}\right) \)
\( M=(1,4) \)
Answer: center \( (1,4) \)
Equation of a Line
A linear equation represents a straight line on the coordinate plane.
The most important part of a line is its slope and a point it passes through.
Slope Formula
\( m=\dfrac{y_2-y_1}{x_2-x_1} \)
Slope tells how steep the line is.
- Positive slope → line rises
- Negative slope → line falls
- Zero slope → horizontal line
- Undefined slope → vertical line
Forms of Equation of a Line
1. Slope–Intercept Form
\( y=mx+c \)
- \( m \) = slope
- \( c \) = y-intercept
2. Point–Slope Form
\( y-y_1=m(x-x_1) \)
Used when slope and one point are known.
Special Lines
- Horizontal line: \( y=k \)
- Vertical line: \( x=c \)
DIGITAL SAT Tip
Most SAT questions give two points. Your job is:
- Find slope
- Substitute into point-slope form
- Rewrite into slope-intercept form
Example 1:
Find the equation of the line through \( (1,2) \) and \( (3,6) \).
▶️ Answer/Explanation
Slope:
\( m=\dfrac{6-2}{3-1}=\dfrac{4}{2}=2 \)
Point–slope form:
\( y-2=2(x-1) \)
Convert to slope–intercept:
\( y-2=2x-2 \)
\( y=2x \)
Answer: \( y=2x \)
Example 2 (Given Slope and Point):
Find the equation of the line with slope 3 passing through \( (2,-1) \).
▶️ Answer/Explanation
\( y+1=3(x-2) \)
\( y+1=3x-6 \)
\( y=3x-7 \)
Answer: \( y=3x-7 \)
Example 3 (Horizontal & Vertical):
Find the equation of a horizontal line passing through \( (4,5) \).
▶️ Answer/Explanation
Horizontal lines have constant y-values.
\( y=5 \)
Answer: \( y=5 \)