1.Without using Pythagorus theorm, prove that the points (-4,-3), (-2,2),(8,-2)are the vertices of a right angled triangle.

2. The 3 vertices of a parallelogram taken in order are (3,4),(-2,3), (-3,-2).Find the coordinates of the fourth vertex ?

1. To prove that the points (-4,-3), (-2,2), and (8,-2) are the vertices of a right-angled triangle without using the Pythagorean theorem, we can instead use the concept of slopes.

First, we calculate the slopes of the line segments formed by the three pairs of points. The slope, denoted by 'm', between two points (x1, y1) and (x2, y2) can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

For the points (-4,-3) and (-2,2):
m1 = (2 - (-3)) / (-2 - (-4)) = 5 / 2

For the points (-2,2) and (8,-2):
m2 = (-2 - 2) / (8 - (-2)) = -4 / 10 = -2 / 5

For the points (-4,-3) and (8,-2):
m3 = (-2 - (-3)) / (8 - (-4)) = -1 / 12

Next, we check if any two slopes are negative reciprocals of each other. In a right-angled triangle, the product of the slopes of perpendicular lines is -1.

Checking the pairs of slopes:

m1 * m2 = (5/2) * (-2/5) = -1

Therefore, the slopes m1 and m2 are negative reciprocals of each other.

Since one pair of slopes are negative reciprocals, we can conclude that the three points (-4,-3), (-2,2), and (8,-2) form a right-angled triangle.

2. The fourth vertex of a parallelogram can be found by using the properties of parallelograms.

A parallelogram is a quadrilateral with opposite sides that are parallel and congruent.

Let the given three vertices of the parallelogram be (3,4), (-2,3), and (-3,-2).

To find the fourth vertex, we need to find the vector from one vertex to another and then translate it to the fourth vertex.

From the given vertices, we can determine the vector from (3,4) to (-2,3):

Vector = (-2 - 3, 3 - 4) = (-5, -1)

Now, we translate the vector by adding it to the third vertex (-3,-2):

Fourth vertex = (-3, -2) + (-5, -1) = (-3 - 5, -2 - 1) = (-8, -3)

Therefore, the coordinates of the fourth vertex are (-8,-3).

1. Show that two of the three line segments have slopes that are negative reciprocals of each other.

2. let the fourth point be (x,y)
Make a sketch to see which lines would be parallel.
You will have 2 equations in 2 unknowns