Identify which lines are parallel.
1. y = 6; y = 6x + 5; y = 6x - 7; y = -8
A: y = 6; y = -8; y = 6x +5; y = 6x - 7 (all)
2. y = 3/4x - 1; y = -2x; y - 3 = 3/4(x - 5); y - 4 = -2(x + 2)
A: y = 3/4x - 1; y- 3 = 3/4(x -5)
Identify which lines are perpendicular.
3. y = 2/3x - 4; y = -3/2x + 2; y = -1; x = 3
A: y = 2/3x - 4; y = -3/2x + 2; y = -1; x = 3 (all)
4. y = -3/7x - 4; y - 4 = -7(x + 2); y - 1 = 1/7(x - 4); y - 7 = 7/3(x - 3)
A: y = -3/7x - 4; y - 4 = -7(x + 2); y - 1 = 1/7(x - 4); y - 7 = 7/3(x - 3) (all)
5. Show that PQRS is a rectangle.
(P: (1,4); Q: (2,6); R: (8,3); S: (7,1)
A: PQ is perpendicular to QR because 2(-1/2) = -1. RS is perpendicular to PS because 2(-1/2) = -1. Therefore. RQRS is a rectangle because it contains a right angle.
on 2, isn't all lines parallel?
on 4, I don't see negative reciprocals.
on 5.
slope PQ = deltay/deltax=-2/-1=2
slope QR= 3/-6=-1/2
slope RS=2/1=2
slope SP=-3/6=-1/2
SP is perpendcular to RS and PQ, and sp is parallel to qr
Reiny?
To identify which lines are parallel, we need to compare their slopes. Lines are parallel if and only if their slopes are equal.
1. The given lines are:
- y = 6
- y = 6x + 5
- y = 6x - 7
- y = -8
To determine if they are parallel, we compare the slopes. The slope of a line in the form "y = mx + b" is represented by the coefficient of x (m).
Comparing the slopes, we find:
- For y = 6: the slope is 0 (no x term)
- For y = 6x + 5: the slope is 6
- For y = 6x - 7: the slope is 6
- For y = -8: the slope is 0 (no x term)
Since the lines with slopes 6 are the same, and the lines with slopes 0 are the same, all four lines are parallel.
Therefore, the answer is: y = 6; y = -8; y = 6x + 5; y = 6x - 7 (all).
2. The given lines are:
- y = 3/4x - 1
- y = -2x
- y - 3 = 3/4(x - 5)
- y - 4 = -2(x + 2)
Again, we need to compare the slopes of the lines to check for parallelism.
Comparing the slopes, we find:
- For y = 3/4x - 1: the slope is 3/4
- For y = -2x: the slope is -2
- For y - 3 = 3/4(x - 5): the slope is 3/4
- For y - 4 = -2(x + 2): the slope is -2
The lines with slopes 3/4 are the same, so they are parallel.
Therefore, the answer is: y = 3/4x - 1; y - 3 = 3/4(x - 5).
To identify which lines are perpendicular, we need to compare their slopes. Lines are perpendicular if and only if the product of their slopes is -1.
3. The given lines are:
- y = 2/3x - 4
- y = -3/2x + 2
- y = -1
- x = 3
Comparing slopes to check for perpendicularity:
- For y = 2/3x - 4: the slope is 2/3
- For y = -3/2x + 2: the slope is -3/2
- For y = -1: the slope is 0 (no x term)
- For x = 3: the slope is undefined (vertical line)
Since the lines with slopes 2/3 and -3/2 have a product of -1, they are perpendicular. The lines with slopes 0 and the vertical line x = 3 are also perpendicular because the product of their slopes is 0.
Therefore, the answer is: y = 2/3x - 4; y = -3/2x + 2; y = -1; x = 3 (all).
4. The given lines are:
- y = -3/7x - 4
- y - 4 = -7(x + 2)
- y - 1 = 1/7(x - 4)
- y - 7 = 7/3(x - 3)
Comparing slopes to check for perpendicularity:
- For y = -3/7x - 4: the slope is -3/7
- For y - 4 = -7(x + 2): the slope is -7
- For y - 1 = 1/7(x - 4): the slope is 1/7
- For y - 7 = 7/3(x - 3): the slope is 7/3
Since the lines with slopes -3/7 and -7 have a product of -1, they are perpendicular. The lines with slopes 1/7 and 7/3 do not have a product of -1, so they are not perpendicular.
Therefore, the answer is: y = -3/7x - 4; y - 4 = -7(x + 2).
Finally, to show that PQRS is a rectangle, we need to verify if its opposite sides are parallel and all angles are right angles.
The given points are:
- P: (1,4)
- Q: (2,6)
- R: (8,3)
- S: (7,1)
To determine if PQ is perpendicular to QR and if RS is perpendicular to PS, we calculate the slopes.
For PQ (line through points P and Q):
- Slope = (y2 - y1) / (x2 - x1) = (6 - 4) / (2 - 1) = 2/1 = 2
For QR (line through points Q and R):
- Slope = (y2 - y1) / (x2 - x1) = (3 - 6) / (8 - 2) = -3/6 = -1/2 = -0.5
The product of the slopes for PQ and QR is equal to -1, so they are perpendicular.
Similarly, we can calculate the slopes for RS (through points R and S) and PS (through points P and S) to determine if they are perpendicular. If the product of their slopes equals -1, then they are perpendicular.
For RS:
- Slope = (y2 - y1) / (x2 - x1) = (1 - 3) / (7 - 8) = -2/(-1) = 2
For PS:
- Slope = (y2 - y1) / (x2 - x1) = (1 - 4) / (7 - 1) = -3/6 = -1/2 = -0.5
The product of the slopes for RS and PS is equal to -1, so they are perpendicular.
Since PQ is perpendicular to QR and RS is perpendicular to PS, we can conclude that PQRS is a rectangle because it contains four right angles.
Therefore, the statement "PQRS is a rectangle" is true.