College Algebra Examples: Applications of the Distance and Midpoint Formulas

Summary

This 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

00:00

📐 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.

05:00

🏀 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.

Keywords

💡Distance Formula

The distance formula is a mathematical formula used to calculate the distance between two points in a coordinate system. It is defined as the square root of the sum of the squares of the differences in the x-coordinates and y-coordinates of the points. In the video, the formula is applied to find the lengths of the sides of a triangle formed by three given points, which is central to understanding the geometric properties of the triangle.

💡Midpoint Formula

The midpoint formula is used to find the midpoint of a line segment given its endpoints. It is calculated as the average of the x-coordinates and the average of the y-coordinates of the endpoints. In the video, this formula is used to determine the position of a shortstop on a baseball diamond, which is a key step in solving the problem related to the distance from the shortstop to home plate.

💡Triangle ABC

In the context of the video, Triangle ABC refers to a geometric figure formed by three points labeled A, B, and C with specific coordinates. The video discusses finding the lengths of the sides of this triangle using the distance formula and verifying if it is a right triangle by applying the Pythagorean theorem.

💡Right Triangle

A right triangle is a type of triangle that has one angle that is exactly 90 degrees. The video script explains how to verify if Triangle ABC is a right triangle by using the relationship between the lengths of its sides, specifically that the sum of the squares of the lengths of the two shorter sides (legs) equals the square of the length of the longest side (hypotenuse).

💡Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. The video uses this theorem to confirm that Triangle ABC is a right triangle by showing that the sum of the squares of the lengths of two sides equals the square of the third side.

💡Area of a Triangle

The area of a triangle can be calculated using the formula 1/2 * base * height. In the video, the area of the right triangle is determined by identifying the base and height after a geometric manipulation that involves 'flipping' the triangle so that the right angle is at the bottom, making it easier to visualize and calculate.

💡Baseball Diamond

A baseball diamond refers to the square-shaped area of a baseball field that includes the four bases: home plate, first base, second base, and third base. In the video, the diamond is used as a context for a problem involving the midpoint and distance formulas, with the shortstop positioned halfway between second and third bases.

💡Coordinate System

A coordinate system is a framework used to describe the position of points in a plane or space. In the video, the coordinate system is used to plot the points that represent the corners of the baseball diamond and to calculate distances and midpoints within this system.

💡Hypotenuse

The hypotenuse is the longest side of a right triangle, opposite the right angle. In the context of the video, identifying the hypotenuse in Triangle ABC is crucial for applying the Pythagorean theorem to verify that the triangle is a right triangle.

💡Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In the video, square roots are used in the distance formula to calculate the lengths of the sides of a triangle and in the area formula for a triangle to find the exact distances and areas without decimal approximations.

💡Exact Form

In mathematics, an exact form of an expression is the form that does not rely on approximations and is expressed in terms of fundamental operations and constants. The video emphasizes leaving answers in their exact form, such as the square root of 20 for the side length of Triangle ABC, instead of using decimal approximations.

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