b The vertices are at each point where two of the three lines intersect.
C
c The sides of the triangle are between the points. How do you calculate the perimeter when you know the sides? How are the base and the height of the triangle defined?
A
a y_1=2x-6 and y_3=- 12x+4 are perpendicular.
B
b (4,2), (2,- 2), and (0,4).
C
c P≈ 15.27 units and A=10 square units.
Practice makes perfect
a Two lines are perpendicular when the product of their slopes is - 1. We have three lines each with a different slope. The lines and the slopes are as follows.
Line
Slope
y_1=2x-6
m_1=2
y_2=- 3x+4
m_2=- 3
y_3=- 1/2x+4
m_3=- 1/2
To find out which two lines that are perpendicular we need to multiply two of the slopes with each other. If the product of the two slopes is - 1 we know that the two lines are perpendicular. There are three combinations and they are
m_1* m_2=2* (- 3)=- 6
m_1* m_3=2* ( - 12 )=- 1
m_2* m_3=- 3* ( - 12 )=- 32.
We now know that the lines y_1=2x-6 and y_3=- 1/2x+4 are perpendicular since the product of their slopes is - 1.
b Each vertex is at a point where two of the three lines intersect. Where two lines intersect they have the same value. We find such points by setting the lines equal and solve the equation. Let's find the x-coordinate for the point where y_1=2x-6 and y_2=- 3x+4 intersect by solving the equation 2x-6=- 3x+4.
The two lines intersect at the coordinate (2,- 2).
Let's now use the same method to find where the lines y_1=2x-6 and y_3=- 1/2x+4 intersect and where y_2=- 3x+4 and y_3=- 1/2x+4 intersect.
y_1 and y_2:& (2,- 2)
y_1 and y_3:& (4,2)
y_2 and y_3:& (0,4)
c We find the perimeter of the triangle by calculating the length of each of the three sides and add them together. Let's name the vertices. Let's call them Q(2,- 2), R(4,2), and S(0,4), where R is a right angle. Let's calculate the length of QR using the Distance Formula.
We continue using the Distance Formula and calculate the lengths of the remaining sides, QS and RS.
QR=sqrt(20)
QS=sqrt(40)
RS=sqrt(20)The perimeter is the sum of the three sides.
The perimeter of the triangle is
P≈ 15.27 units.
When we calculate the area of a triangle we use the formula
A=1/2bh
where b and h are perpendicular to each other. Since we have a right angle in the triangle, we know that the two sides that meet there, QR and RS, are perpendicular with each other. We can, therefore, define that
b=QR and h=RS.
When we calculated the triangle's perimeter we learned the lengths of each of the sides. The side QR has the length sqrt(20), therefore
b=sqrt(20) units.
The side RS is also sqrt(20) units long making the height also having the length
h=sqrt(20) units.
The triangle's area is
A&=1/2bh
&=1/2* sqrt(20) * sqrt(20)
&=10 square units.
