IB Math Studies – Straight lines

IB Math Studies – Straight lines

1. IB Mathematics Studies, Distance between two points

How can we find the distance between two points A(12,4) and B(-10,8)?

Solution

IB Mathematical Studies, Coordinate geometry

The distance between two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given by the following formula:

d_{AB}= \sqrt{( x_{2}- x_{1})^2+( y_{2}- y_{1})^2}

Applying the above formula in our case we have

d_{AB}= \sqrt{(-10-12)^2+(8-4)^2}=\sqrt{484+16}=\sqrt{500}=10\sqrt{5}

2. IB Mathematical Studies, Coordinate geometry

How can we find the parameter c if these A(4,c) and B(-2,3) are 6 units apart?

Solution

IB Maths Studies, Coordinate geometry, Distance Formula

The distance between two points A(x_{1}, y_{1}) and

B(x_{2}, y_{2}) is given by the following formula:

d_{AB}= \sqrt{( x_{2}- x_{1})^2+( y_{2}- y_{1})^2}

Applying the Distance formula in your case we have

d_{AB}= \sqrt{(-2-4)^2+(3-c)^2}=\sqrt{36+(3-c)^2}=6 \rightarrow

\rightarrow ( \sqrt{36+(3-c)^2})^2=6^2 \rightarrow (by squaring both sides)

\rightarrow (36+(3-c)^2)=36 => (3-c)^2=0 \rightarrow 3-c=0

\rightarrow c=3

3. IB Math Studies, Coordinate geometry, Gradient of a line

How can we find the slope of the line passes through these points A(-3,6) and B(2,5)?

Solution

IB Math Studies, Coordinate geometry, Gradient of a line

the slope m of the line passes through two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given by the following formula:

m=\frac{ y_{2}- y_{1}}{ x_{2}- x_{1}}

Applying the above formula we can find the slope of the line

m=\frac{5-6}{2-(-3)}= -\frac{1}{5}

4. IB Math Studies, slope of a line, perpendicular and parallel lines

How can we find the slope of the line which is perpendicular to the line y=-5x+14 ?

Solution

IB Mathematical Studies, gradient of a line, perpendicular and parallel lines

If two lines are perpendicular their slopes are negative reciprocals and if two lines are parallel have equal slopes.

Regarding your question about the gradient of the line which is perpendicular to the line y=-5x+14,

Let m be the required slope then we have the following relation

m \cdot (-5)=-1 =>m=\frac{-1}{-5}= \frac{1}{5}

5. IB Math studies, midpoint formula

How can we find the coordinates of the midpoint of line segment AB where A(1,-5) and B(12,-3) ?

Solution

Midpoint formula, IB Math studies

The coordinates of the midpoint of a line segment defined by the points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given by the following formula:

(x_{m}, y_{m} )=( \frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2} )

Applying the midpoint formula in our case we have the following:

(x_{m}, y_{m} )=( \frac{1+12}{2},\frac{-5-3}{2} )=

=( \frac{13}{2},\frac{-8}{2} )

=(\frac{13}{2}, -4)

Midpoint of a Segment

 

The Distance Formula

 

 Find the Slope Given Two Points and Describe the Line