I'm very confused with what I'm learning about right now. We are kind of talking about slopes, functions, graphs, etc. Today, my teacher taught about midpoint and distance, and did not explain very well at all. Would someone please explain to me how to find midpoint and distance? I have formulas, but I have no idea how to use them, and when I try them, I get the wrong answer.
Here's one of the questions I must answer:
Use slope to verify that your point from part a (which was line segment PQ, with P being (2, 4) and Q being (-7, -2)) is on the line through P and Q, and use the distance formula to show that your point is just as far from P as it is from Q.
Thank you so much! This homework will be collected tomorrow, so I really need to be able to understand this and apply what you tell me to other problems. Thanks!
Sorry, what I wrote might not have made sense. Here's a re-written version.
I'm very confused with what I'm learning about right now. We are kind of talking about slopes, functions, graphs, etc. Today, my teacher taught about midpoint and distance, and did not explain very well at all. Would someone please explain to me how to find midpoint and distance? I have formulas, but I have no idea how to use them, and when I try them, I get the wrong answer.
Here's one of the questions I must answer:
Use slope to verify that your midpoint from part a (which was the midpoint of line segment PQ, with P being (2, 4) and Q being (-7, -2)) is on the line through P and Q, and use the distance formula to show that your midpoint is just as far from P as it is from Q.
Thank you so much! This homework will be collected tomorrow, so I really need to be able to understand this and apply what you tell me to other problems. Thanks!
Well, first of all let's find a point halfway between P and Q.
That point will have an x value halfway between 2 and -7
That is (2-7)/2 = -5/2 = -2.5
The vy value of that halfway point will be halfway between 4 and -2
That is (4-2)/2 = 2/2 = 1
so our halfway point is
( -2.5 , 1 )
now they ask us to verify that
well what is the distance squared from
(2 , 4) to (-2.5 , 1) ?
x difference = -2.5 -2 = -4.5
y difference = 1 - 4 = -3
square of hypotenuse = 4.5^2+3^2 = 29.25
Now do the same thing from
(-2.5 , 1) to (-7 , -2)
x difference = -7 + 2.5 = -4.5
y difference = -2 -1 = -3
square of hypotenuse = same
done
To make sure you have the correct numerical answers, take the square root of 29.25 for your actual half distance.
I kind of get what you're saying... but what is a hypotenuse?
Thanks for the help!
http://www.mathwarehouse.com/geometry/triangles/right-triangle.html
Thank you!
I can help you understand how to find the midpoint and distance between two points.
To find the midpoint of a line segment defined by two points, you can use the midpoint formula:
Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2)
For the given line segment PQ, with P(2, 4) and Q(-7, -2), we can use the midpoint formula to find the coordinates of the midpoint.
Midpoint: ((2 + -7)/2, (4 + -2)/2)
= (-5/2, 2/2)
= (-5/2, 1)
So, the midpoint of line segment PQ is (-5/2, 1).
To verify that this midpoint lies on the line through points P and Q, we can calculate the slope of PQ and compare it to the slope of the line passing through P, Q, and the midpoint.
The slope of a line can be found using the slope formula:
Slope formula: (y₂ - y₁)/(x₂ - x₁)
Let's calculate the slope of PQ first:
Slope of PQ: (-2 - 4)/(-7 - 2)
= -6/-9
= 2/3
Now, let's calculate the slope of the line passing through P, Q, and the midpoint (-5/2, 1):
Slope of line P-Midpoint: (1 - 4)/((-5/2) - 2)
= -3/((-5/2) - 4)
= -3/((-5/2) - (8/2))
= -3/((-5 - 8)/2)
= -3/(-13/2)
= -3 * (2/-13)
= 6/13
So, the slopes of PQ and the line P-Midpoint are not equal (-3/(-13/2) ≠ 2/3). Therefore, the midpoint does not lie on the line through points P and Q.
Moving on to the second part of the question, to show that the midpoint is equidistant from points P and Q, we can use the distance formula.
Distance formula: √((x₂ - x₁)² + (y₂ - y₁)²)
Let's calculate the distance from the midpoint (-5/2, 1) to P(2, 4):
Distance from midpoint to P: √((2 - (-5/2))² + (4 - 1)²)
= √((2 + 5/2)² + 3²)
= √((4/2 + 5/2)² + 3²)
= √((9/2)² + 3²)
= √((81/4) + 9)
= √(81/4 + 36/4)
= √(117/4)
= √117/2
= √(117/2) * 2/2
= √(117*2)/2
= √234/2
Similarly, we can calculate the distance from the midpoint to Q(-7, -2). You will find that it comes out to √234/2 as well.
Therefore, the distance from the midpoint to both P and Q is the same, which confirms that the midpoint is equidistant from P and Q.
I hope this explanation helps you understand how to find the midpoint and distance between two points using the formulas. Let me know if you have any further questions!