<DOE contains points D(2,3) O(0,0) and E(5,1). Which of the following angles are both adjacent and supplementary to <DOE?
<DOF, with F (10, 2)
<DOF, with F (-2, -3)
<DOF, with F (-5, -1)
<EOF, with F (-5, -1)
<EOF, with F (-2, -3)
<EOF, with F (10, 2)
You can choose 2 options!
I'm not good with this stuff. I think one is -5, -1 and the other is -2, -3... maybe. Even so, I wouldn't know if it's DOF or EOF. Please help? Thanks!
First step: PLOT THE POINTS!
DOF must form a straight line with OE, which is the line y=x/5
So, just look at the points where that holds, and decide which angles are adjacent to DOE.
Hint. <DOF, with F (10, 2) does not work, because that is just the same angle as DOE, not adjacent to it.
Thanks, Steve!:) I got the answer!
C and E (if I remember correctly that's what I put down XD) But I got it right:)
To find the angles that are both adjacent and supplementary to <DOE, we need to first calculate the slope of line DE and then use it to determine the slopes of the other lines.
The slope of line DE can be found using the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (2, 3) and (x2, y2) = (5, 1).
m = (1 - 3) / (5 - 2)
m = -2 / 3
Now, let's find the slopes of the other lines:
Line DOF, with F(10, 2):
m = (2 - 3) / (10 - 2)
m = -1 / 8
Line DOF, with F(-2, -3):
m = (-3 - 1) / (-2 - 5)
m = -4 / -7
m = 4 / 7
Line DOF, with F(-5, -1):
m = (-1 - 1) / (-5 - 2)
m = -2 / -7
m = 2 / 7
Line EOF, with F(-5, -1):
m = (-1 - 3) / (-5 - 2)
m = -4 / -7
m = 4 / 7
Line EOF, with F(-2, -3):
m = (-3 - 1) / (-2 - 5)
m = -4 / -7
m = 4 / 7
Line EOF, with F(10, 2):
m = (2 - 3) / (10 - 2)
m = -1 / 8
Now, we need to check which lines have slopes that are both adjacent and supplementary to the slope of line DE (-2/3).
Adjacent angles have slopes whose absolute difference from the slope of line DE is 1. Supplementary angles have slopes whose sum with the slope of line DE is 0.
By checking the slopes, we can see that:
- The slope of line DOF, with F(-2, -3) is adjacent (-4/7) but not supplementary.
- The slope of line DOF, with F(-5, -1) is adjacent (2/7) and supplementary.
- The slope of line EOF, with F(-5, -1) is adjacent (4/7) and supplementary.
- The slope of line EOF, with F(-2, -3) is adjacent (4/7) but not supplementary.
So, the two angles that are both adjacent and supplementary to <DOE are:
<DOF, with F(-5, -1)
<EOF, with F(-5, -1)
To determine which angles are both adjacent and supplementary to <DOE, we need to find the slope of line segment DE and use it to find the equations of the lines containing segments DE and EF. Then, we can find the angles formed by these lines and determine which angles are adjacent and supplementary to <DOE.
Step 1: Find the slope of line segment DE
The slope of a line through two points (x1, y1) and (x2, y2) is given by the formula: m = (y2 - y1) / (x2 - x1).
The slope of DE = (y2 - y1) / (x2 - x1)
= (1 - 3) / (5 - 2)
= (-2) / 3
Step 2: Find the equation of line DE
The equation of a line with slope m and passing through a point (x1, y1) is given by the point-slope form: y - y1 = m(x - x1).
Using the slope (-2/3) and the point D(2, 3), the equation of line DE is:
y - 3 = (-2/3)(x - 2)
y - 3 = (-2/3)x + 4/3
y = (-2/3)x + 13/3
Step 3: Find the equation of line EF
Note that we have three options for the point F: F(10, 2), F(-2, -3), and F(-5, -1). We need to find the line that passes through point F and is also perpendicular to line DE.
Since the slopes of perpendicular lines are negative reciprocals of each other, the slope of the line EF should be 3/2.
For F(10, 2):
y - 2 = (3/2)(x - 10)
y - 2 = (3/2)x - 15/2
y = (3/2)x - 11/2
For F(-2, -3):
y - (-3) = (3/2)(x - (-2))
y + 3 = (3/2)x + 3
y = (3/2)x
For F(-5, -1):
y - (-1) = (3/2)(x - (-5))
y + 1 = (3/2)x + 15/2
y = (3/2)x + 13/2
Step 4: Find the angles and check for adjacency and supplementary
To find the angles, we need to find the angle between lines DE and EF. The formula for the angle between two lines with slopes m1 and m2 is given by: angle = arctan((m2 - m1) / (1 + m1 * m2)).
Using the slopes -2/3 (for DE) and 3/2 (for EF):
For F(10, 2):
angle DOF = arctan((3/2 - (-2/3)) / (1 + (-2/3)(3/2)))
For F(-2, -3):
angle DOF = arctan((3/2 - (-2/3)) / (1 + (-2/3)(3/2)))
For F(-5, -1):
angle EOF = arctan((3/2 - (-2/3)) / (1 + (-2/3)(3/2)))
To determine which angles are adjacent and supplementary to <DOE, you can calculate the angles using the above formulas and check if they are adjacent (share a common side) and supplementary (sum of angles is 180 degrees).
Oh, geometry! Time to put on my clown nose and get silly with angles!
Okay, let's figure this out together. We're looking for angles that are both adjacent and supplementary to <DOE. Adjacent means they share a common side, and supplementary means they add up to 180 degrees.
Let's check the options one by one:
1. <DOF, with F (10, 2): Hmm, this angle is not adjacent to <DOE. Unless the angle does some teleporting tricks, this option is out.
2. <DOF, with F (-2, -3): Yikes, another non-adjacent option. Looks like this angle and <DOE need to expand their social circle.
3. <DOF, with F (-5, -1): Ah, finally an adjacent friend! But wait, are they supplementary? Let's see, if I add <DOE and <DOF, do they make a complete 180 degrees? Hmmm, I'm afraid not. They don't add up to a full circle, so this option won't work either.
4. <EOF, with F (-5, -1): Oh look, another potential friend! They're adjacent to <DOE, but are they supplementary? Adding them up, I'm afraid they don't cut it. They're not a dynamic duo of 180 degrees.
5. <EOF, with F (-2, -3): Oh hi there, adjacent angle! But again, the sum of the angles isn't quite right. They're not supplementary, so we'll have to say goodbye.
6. <EOF, with F (10, 2): Hey, someone who's adjacent to <DOE again! But let's do some math. Adding up the angles, they do make 180 degrees! Hooray, we've finally found a winner!
So, to recap, the angles that are both adjacent and supplementary to <DOE are <EOF with F (-5, -1) and <EOF with F (10, 2). We've cracked the code, my friend! You're doing great, keep up the angle adventures!