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 ?
can u pl explain me in detail.
how far have you gotten using Reiny's suggestion?
For #2, you might also look at it as involving the translation of one line onto a parallel line.
The line from (-2,3) to (3,4) is one side of the parallelogram.
If you translate (-2,3) to (-3,-2) then you have moved it by (-1,-5). So, move the other end of the line by the same amount, and you will have found the other end of the parallel side.
Sure! I'd be happy to explain in detail.
1. To prove that the points (-4, -3), (-2, 2), (8, -2) are the vertices of a right-angled triangle without using Pythagoras' theorem, we can make use of the concept of perpendicular slopes.
First, let's find the slopes of the two line segments formed by the three points.
The slope of the line passing through the points (-4, -3) and (-2, 2) can be calculated using the formula: m = (y2 - y1) / (x2 - x1). Substituting the values, we get:
m1 = (2 - (-3)) / (-2 - (-4)) = 5 / 2.
Similarly, the slope of the line passing through the points (-2, 2) and (8, -2) can be calculated as:
m2 = (-2 - 2) / (8 - (-2)) = -4 / 10 = -2 / 5.
Now, to determine if these two line segments are perpendicular to each other, we need to check if their slopes are negative reciprocals of each other. That is, m1 * m2 = -1.
Multiplying the two slopes together, we get:
(5 / 2) * (-2 / 5) = -1.
Since the product of the slopes is -1, we can conclude that the line segments formed by the three points (-4, -3), (-2, 2), and (8, -2) are perpendicular to each other. Therefore, these points form a right-angled triangle.
2. To find the coordinates of the fourth vertex of the parallelogram, we can use the concept of opposite sides being parallel and equal.
Let's consider the given vertices: (3, 4), (-2, 3), and (-3, -2). To find the fourth vertex, we calculate the change in the x-coordinate and the change in the y-coordinate between the first two vertices.
Change in x-coordinate: (-2 - 3) = -5
Change in y-coordinate: (3 - 4) = -1
Now, to find the fourth vertex, starting from the third vertex (-3, -2), we add the change in x-coordinate and the change in y-coordinate:
X-coordinate of the fourth vertex = -3 + (-5) = -8
Y-coordinate of the fourth vertex = -2 + (-1) = -3
So, the coordinates of the fourth vertex are (-8, -3).
Therefore, the fourth vertex of the parallelogram is (-8, -3).