College Algebra Examples: Applications of the Distance and Midpoint Formulas
Summary
TLDRThis educational video script explores the application of distance and midpoint formulas through geometric examples. It begins with a problem involving a triangle, calculating side lengths and confirming it's a right triangle using the Pythagorean theorem. The script then demonstrates calculating the area of the triangle using geometric manipulation. The second example applies these formulas to a baseball scenario, determining the distance a shortstop must throw to home plate. The script effectively illustrates the practical use of mathematical concepts in real-world situations.
Takeaways
- 📚 The video discusses the application of distance and midpoint formulas through examples.
- 📐 The first example involves finding the lengths of the sides of a triangle with given vertex coordinates.
- 📈 The distance formula is used to calculate the lengths of the sides of the triangle ABC.
- 🔍 The coordinates of points A, B, and C are given in terms of a variable 'a', which is used in calculations.
- 📉 The lengths of the sides are found to be the square root of 20, 5, and the square root of 5 respectively.
- 🔺 Part B of the problem asks to verify if the triangle is a right triangle using the Pythagorean theorem.
- 📐 The triangle is confirmed to be a right triangle as the sum of the squares of the two shorter sides equals the square of the longest side.
- 📏 The area of the right triangle is calculated using the formula (1/2) * base * height, resulting in an area of 5.
- 🏀 The second problem involves a baseball diamond and a short stop throwing to home plate.
- 📍 The baseball diamond is modeled as a square with home plate at the origin and the bases at specific coordinates.
- 🤸♂️ The midpoint formula is used to find the position of the short stop, who is halfway between second and third base.
- 📏 The distance from the short stop to home plate is calculated using the distance formula, resulting in approximately 100.6 feet.
Q & A
What are the coordinates of point A in the first example problem?
-The coordinates of point A are (-2a, 1).
What is the formula used to calculate the distance between two points?
-The distance formula is used, which is the square root of the difference in the x-coordinates squared plus the difference in the y-coordinates squared.
How is the length of each side of triangle ABC found?
-The length of each side is found by applying the distance formula to the coordinates of the respective points.
What is the length of side AB in triangle ABC?
-The length of side AB is the square root of 20.
How can you verify if a triangle is a right triangle?
-You can verify if a triangle is a right triangle by checking if the sum of the squares of the lengths of the two shorter sides equals the square of the length of the longest side (the hypotenuse).
What is the relationship between the sides of a right triangle in terms of their lengths?
-In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.
What is the area of the right triangle ABC?
-The area of the right triangle ABC is 5 square units, calculated by taking half the product of the lengths of the two legs (base and height).
How is the midpoint of a segment found?
-The midpoint of a segment is found by averaging the x-coordinates and the y-coordinates of the endpoints of the segment.
What is the scenario described in the second problem involving a baseball diamond?
-The second problem involves a shortstop standing halfway between second and third base on a baseball diamond and needing to throw the ball to home plate, with the goal of determining the distance from the shortstop to home plate.
How is the distance from the shortstop to home plate calculated in the second problem?
-The distance is calculated using the distance formula with the coordinates of the shortstop (45, 90) and home plate (0, 0).
What is the approximate distance the shortstop has to throw the ball to reach home plate?
-The shortstop has to throw the ball approximately 100.6 feet to reach home plate.
Outlines
📐 Applications of Distance and Midpoint Formulas
This paragraph introduces the topic of the video, which is the application of distance and midpoint formulas through examples. The first example involves finding the lengths of the sides of a triangle with given vertex coordinates and then verifying if the triangle is a right triangle using the Pythagorean theorem. The process includes plotting the points, applying the distance formula, and checking the relationship between the sides' lengths. The video also covers calculating the area of the right triangle by considering geometric manipulation.
🏀 Baseball Diamond Problem Using Midpoint and Distance Formulas
The second paragraph presents a problem involving a baseball diamond where the goal is to determine the distance a shortstop needs to throw a ball to reach home plate from a position exactly halfway between second and third bases. The solution involves assigning coordinates to the bases, calculating the midpoint where the shortstop stands, and then using the distance formula to find the distance from this midpoint to home plate. The problem demonstrates the practical application of the midpoint formula to find a position and the distance formula to calculate the required throw distance, concluding with an approximate numerical answer.
Mindmap
Keywords
💡Distance Formula
💡Midpoint Formula
💡Triangle ABC
💡Right Triangle
💡Pythagorean Theorem
💡Area of a Triangle
💡Baseball Diamond
💡Coordinate System
💡Hypotenuse
💡Square Root
💡Exact Form
Highlights
Introduction to applications of distance and midpoint formulas with examples.
Problem setup involving points A, B, and C with variable coordinates based on 'a'.
Plotting points on a coordinate system to visualize triangle ABC.
Using the distance formula to calculate the length of sides of triangle ABC.
Calculation of the distance from A to B using coordinates and the distance formula.
Explanation of leaving answers in exact form instead of decimal approximations.
Finding lengths of other two sides of the triangle using the distance formula.
Verification of triangle ABC being a right triangle using the Pythagorean theorem.
Graphical reasoning for the likely location of the right angle in triangle ABC.
Mathematical confirmation of the right angle using the lengths of triangle sides.
Determining the area of the right triangle using base and height.
Geometric manipulation to find the area of triangle ABC as 5 square units.
Introduction to a second problem involving a baseball diamond and a shortstop.
Setting up a coordinate system with home plate at the origin for the baseball diamond.
Determining the coordinates of second and third bases and the midpoint for the shortstop's position.
Using the midpoint formula to find the exact location of the shortstop.
Applying the distance formula to calculate the throw distance from the shortstop to home plate.
Final calculation showing the shortstop needs to throw the ball approximately 100.6 feet.
Transcripts
in this video we'll learn about some
applications of the distance and
midpoint formulas through some
examples so let's look at our first
example problem consider the three
points a which has coordinates -2a 1 B
which has coordinates 2A 3 and C which
has coordinates 3A 1 so first thing they
ask us to do is find the length of each
side of the triangle ABC now it's going
to be helpful here for us to have a
picture so let's just plot these three
points really quick so I'm going to draw
my axes here so -2 comma 1 I'm just
going to take off some points on my Axis
so -2 comma 1 is going to be this point
right here -2 on the xaxis one on the Y
AIS there's my point a 2 comma 3 is
going to be over here two on the x-
axxis three on my y- AIS and then 3
comma 1 that's going to be over here 3
on the xaxis one on the y- AIS so my
triangle is going to be something like
this now if I want to find the length of
each side of the triangle I'm going to
use the distance formula so the distance
from A to B is going to be the square
root of the difference in the x
coordinates squared plus the difference
in the y-coordinates squared so the x
coordinate of a is
-2 the x coordinate of B is 2 so -2
minus 2 and then I'm going to square
that plus and then the uh difference in
the y-coordinates the Y coordinate of a
is 1 and the y coordinate of B is 3 so I
have 1 - 3^ 2 and that's it we just plug
into the distance formula and then see
what this works out to be so this is
going to be
-4 2ar and then 1 - 3 is -2 squar -4 2
is POS
16 -2 2 is pos4 so that works out to be
the < TK of
20 and in general you should leave your
answer in exact form whenever possible
we could type the square root of 20 into
our calculators and get a decimal
approximation and that might sometimes
be useful but the exact answer here is
the square root of
20 similarly we can find the lengths of
the other two sides of this triangle so
the distance from a to
c is going to work out to be five and
the distance from B to
C is going to work out to be the square
root of 5
so check those on your own and make sure
that you can get those answers but those
are our other two distances that we're
looking
for now for Part B of this problem when
they ask us to verify that the triangle
is a right triangle what exactly are
they asking us to do well remember that
when we have a right triangle we have
the relationship that a s+ BAL c^2 the
sum of the squares of the lengths of the
legs equals the square of the length of
hypotenuse but the first problem here is
that we're May not exactly sure which of
these three sides is is supposed to be
the hypotenuse or in other words we
don't know where the right angle is
supposed to be but if we look at our
picture and this is one of the good
reasons uh that we wanted to write draw
a picture here it it looks like if
there's going to be a right angle in
this triangle it looks like it might be
there now we don't know that that's a
right angle yet but if we had to guess
that there would be one then that's
where we would guess it would be just by
looking at it that looks much more like
a right angle than the other two angles
in this triangle so what would have to
be true about these three distances in
order for this to be a right triangle
well what have we figured out we figured
out that this distance is theun of
20 we figured out that the distance from
B to C is the square otk of 5 and we
figured out that the distance from a to
c is just five so what would have to be
true here is that the distance Square <
TK of
20 squar plus theun of 5 squar would
have to equal we don't know if it does
yet but it would have to equal 5^
s well theot of 20 s when I square a
square root the square root goes away I
just get 20 again when I square a square
root the square root goes away and I
just get five so does 20 + 5 equal 5^ s
well 5 squ is 25 so looks like that
works so now we know that the Triangle
really is a right triangle so what's the
area of that right triangle well the
area is 12 B base time
height now it might be a little bit
tricky to think about what the height of
this triangle is if we were trying to
look at this triangle the way it's
oriented on the screen right now we're
trying to figure out the height of this
triangle in other words this distance
here that I'm drawing in pink that pink
height would actually be quite difficult
to figure out given the information that
we have but what we can do instead is we
can flip the triangle
around and draw it so that the right
angle is at the bottom
if we do that then what we know is that
the base of my triangle which is the
short side that's theun of 5 and the
height of my triangle that's this tall
side that's theare of 20 so now I know
the base and the height of my triangle
now that I flipped it around this way so
I get 12 the base of my triangle is
theare of 5 the height of my triangle is
theare of 20 and when I multiply those
together if I multiply two square roots
the product of two square roots is the
square root of a product so that's 5 *
20 under the root which is
100 the square root of 100 is 10 and so
I get 12 of 10 which works out to be
five so we've got all the answers to our
questions here we found the lengths of
the three sides of our triangle we found
that it really is a right triangle and
then we've used that with a little bit
of geometric manipulation uh to figure
out that the area of the triangle is
five
all right let's do another problem this
time we're going to use both the
midpoint and the distance formulas so
we're told that a baseball diamond is a
square 90 ft on a side and that a short
stop is standing exactly halfway between
second and third base and needs to throw
the ball to home plate and we want to
know how far from home plate is he so
the way we're going to do this we want
to use our knowledge of our coordinate
system and so far it doesn't look like
this problem has anything to do with
coordinates so let's make it have
something to do with coordinates let's
draw our baseball diamond so that home
plate is right here at the origin so
we'll call that
home first base will be along the xaxis
so we'll call that
first second base is going to be out
here up in quadrant one so call that
second and then Third Base will be over
here on the Y
AIS and so our baseball diamond we can
see is the square right
here so because the baseball diamond is
90 ft on a side that's 90 that's 90
that's 90 and that's
90 then we can figure out the
coordinates of these points this point
is going to be the point 90 comma 0
because we're 90 spaces over on the
xaxis second base is going to be 90
comma
90 and then third base is going to be uh
0 comma
90 so when we're told that we're going
to be exactly halfway between second and
third base that means that this green
point right here that's where our Short
Stop is that's where our little short
stop here is so you appreciate my lovely
art so where is that green Point what
are the coordinates of that green Point
well the
midpoint is given by remember the
average of the x coordinates in this
case that's 0 +
90 / 2 comma the average of the
y-coordinates which in this case is 90 +
90 / 2 so I added the two x coordinates
together the 0 and the 90 and that gave
me my midpoint x coordinate and I'm
adding my two Y coordinates 90 and 90
and averaging those and that's going to
give me the y coordinate of my midpoint
so when I work that
out that's going to work out to be
45 comma
90 so that's where our Short Stop is
standing so now what we want to know
what the question is asking us for is
what's the distance between the short
stop and home plate but we know the two
coordinates of those points now we know
that the short stop is standing at 45a
90 and we know that home plate is at the
origin 0 comma 0 so now we're going to
use our distance formula and distance
formula tells us that the distance that
the short stop has to throw the ball is
the square root of the difference in the
x coordinates remember home plate is at
0 0 so the x coordinates difference is
45 -
0
squar Plus 90 - 0
squ now the numbers are getting a little
big so we'd like to uh use our
calculator
here 45 - 0 is 45 and when we square
that we get
225 90 - 0 is 90 and when we square that
we get
8100 adding together 2,25 and 8100 gives
us
10,125 and when we compute the sare root
of 10,125 on our calculator we get that
it's approximately equal to
100.6 so that means that our Short Stop
has to throw the
ball just over
100t to get to from his position to home
plate and that's our answer
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