a To find the equation of CM, we have to determine it's slope, m, and y-intercept b.
y= mx+ b
Examining the graph, we can identify where CM intercepts the y-axis.
The line intercepts the y-axis at (0,2) which means b= 2. To find the slope, we have to measure the vertical and horizontal distance between two points on the graph. Since both C and M are points with integer coordinates, let's use these to identify the slope.
Traveling from the C to M requires that we move 4 steps horizontally and 2 steps vertically, both in the positive direction. With this, we can find the slope.
rise/run=2/4 ⇔ m= 1/2
Now that we have the slope and y-intercept, we can write our final equation.
y= mx+ b
y= 1/2x+ 2
b Let's highlight △ CPM in the diagram. Note that one leg is horizontal and the other is vertical. This means the legs are perpendicular and △ CPM must be a right triangle.
The area of a triangle is its base multiplied by its height divided by 2.
A=1/2bh
Since this is a right triangle, it's legs will make up the triangle's height and base. Let's mark the lengths of these sides in the diagram.
With this, we have all the information we need to calculate the area of the triangle.
The area of the triangle is 4. To find the perimeter of the triangle, we have to add all of its sides. Since we do not know the hypotenuse, we have to calculate it. We can do this by using the Distance Formula.
When we know the hypotenuse, we can finally calculate the perimeter by adding all of the sides.
Perimeter: 4+2+sqrt(20)≈ 10.47
c To write the equation of a line perpendicular to CM and through point M, we first need to determine its slope. Two lines are perpendicular when their slopes are opposite reciprocals. This means that the product of a given slope and the slope of a line perpendicular to it will be -1.
m_1* m_2=-1
From Part A, we know the slope of CM. By substituting the slope into the equation, we can solve for the slope of the perpendicular line, m_2.
Any line perpendicular to the given equation will have a slope of - 2 which means we can write the equation in the following form.
y= - 2x+b
To find b, we substitute the coordinates of point M in the equation and solve for b.
