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in this video we're going to talk about

how to find the area

of an isosceles triangle

now keep in mind an isosceles triangle

has

two side lengths that are the same

they're of equal length

now in order to calculate the area

of any triangle

there's two things that we need to know

we need to know the base of the triangle

and the height of the triangle

if we could find those two pieces of of

information we could use this formula

area is one half base times height

now we have the base but we don't know

the value of the height so we got to

find that

what I'm going to do is I'm going to

break this triangle into two equal parts

but I'm only going to draw the left side

of that of what remains

so here it is

once you do that this becomes a right

triangle

and in this right triangle we have a

height of H

we only have one half the base now so

this is going to be eight

and this part is still 17.

because we have a right triangle

we could use the Pythagorean theorem to

calculate the value of H

so it's going to be a squared plus b

squared is equal to c squared

now let's make a equal to H

I mean a equal to 8 Rather B is going to

be the height h

the hypotenuse which is across the right

angle that's C that's going to be the

longest side of the triangle which is

17.

so now let's plug in what we have

8 squared plus

B is going to be H so this is H squared

and C is 17.

now 8 squared

8 squared is 8 times 8 which is 64.

17 squared I'm going to have to

use a calculator for that

17 times 17 is 289.

now to get H squared by itself

we need to subtract both sides by 64.

and we can bring down

the H squared

so 289 minus 64 is 200 and 225.

now to get H by itself we can take the

square root of both sides

and the square root of 225 is 15.

so the height of the triangle is 15

units

now let's redraw the original triangle

but only with the relevant parts that we

need

so the base is still 16.

but the height

we know the height to be

15.

this is the 8 15 17 triangle

so now we can calculate the area

now that we know the values of B and H

so it's one half times the base which is

16. times the height which is 15.

half of 16 is 8.

and 8 times 15

. if you want to do it mentally you can

write it this way

15 is 10 plus 5.

8 times 10 is 80.

8 times 5 is 40.

so this is going to give us 120.

so that is the area

of this particular isosceles triangle

it's 120 square units

now let's talk about another way in

which we can get the answer

anytime you know all three sides

of a triangle you could use huron's

formula

and here it is first

you need to calculate s which is a plus

b plus c

divided by 2.

now a b and c those are the three side

lengths of the triangle so we can say

that a is 16.

B is 17 and C is 17.

now once you have S you can calculate

the area using this formula it's the

square root of s

times s minus a times s minus B

times s minus C so what we're going to

do is we're going to use this formula or

these two formulas rather to confirm our

answer to make sure that this answer is

correct

so let's plug in the three side lamps

into the first equation

so it's going to be 16 plus 17 plus 17.

divided by 2.

16 plus 17.

that's 33

and 33

well let's do it this way let's add 17

plus 17 first

17 plus 17 is 34.

and 16 plus 34 is 50.

half of 50 is 25.

so our s value

is

25.

now that we know the value of x we can

use this formula to get the area

so we have the square root of 25

and then it's 25 minus a a is 16.

25 minus B we know B is 17 and so is C

now 25 minus 16 that's going to be 9.

25 minus 17 is 8.

and we have two of them

so now let's do some

simplifying

now I'm going to break this into three

parts first we have 25 which is a

perfect square so I'm going to leave it

by itself in a radical then we have the

square root of 9.

now we can't take the square root of

eight but 64 we can 8 times 8 is 64.

so this is what we now have

the square root of 25 is 5. the square

root of 9 is 3 and the square root of 64

is 8.

now 5 times 3 is 15. and we already know

what 15 times 8 is is the same as 8

times 15 which is 120.

so we do get the same answer

regardless of the methods that we choose

to use

so the area of this particular isosceles

triangle is 120 square units

now for the sake of practice

go ahead and try this problem

calculate the area of this isosceles

triangle feel free to pause the video

and work on this example

so we're going to use the first method

because it's a lot faster

so let's draw half

of this triangle

and let's calculate the height

so the base is going to be half of what

it was which is 7. but the hypotenuse is

still going to be 25.

now at this point it helps to know the

special right triangles

for instance there's the three four five

a right triangle

there's the 5 12 13 right triangle

there's the 8 15 17 triangle

and then there's the 724

25 triangle which is what we have so

knowing this you can automatically know

that H is 24. now there's some other

ones like the 940 41

and I believe 11 60 61. let me make sure

that's correct

and it is

for the most part

you'll have one of those six variants or

any multiple of them so for instance if

you multiply this by two you can have

the six eight

ten triangle or if you multiply it by

three then 9 12 15 triangle

so any multiples of those numbers will

work as well

so like this one if you multiply it by

two you have the 10 24 26 triangle

so now that we know the height of the

triangle

let's redraw the picture

so the base is 14.

and the height

is 24.

so we can go ahead and use this formula

the area is going to be one half base

times height

base is 14. and the height is 24.

one half of 14 is seven and so this

becomes 7 times 24.

so to do it mentally let's break down

24 into 20 plus 4.

7 times 20 if you have seven twenty

dollar bills how much money do you have

that's going to be 140.

and then 7 times 4 is 28. adding those

two together

we get 168.

so that's the area of the triangle

and let's confirm it real quick using

Heron's formula

so let's calculate s

is going to be one half the sum of the

three sides

so 14 plus 25 plus 25 divided by 2.

25 plus 25 is 50 50 plus 14 is 64.

half of 64 is 32. so we could say s is

32.

now the area is going to be the square

root of s

and then s minus a

let's say 32 minus 14 that's 18. and

then s minus B 32 minus 25 that's 7 and

then 32 minus the other 25 is 7 as well

32 we can write that as 16 times 2.

18 that's 2 times 9.

and 7 times 7 I'm going to leave it as

49. I'm trying to collect all the

perfect squares here

so 16 is the perfect square 2 times 2 is

4. we have 9 and 49.

the square root of 16 is 4 the square

root of 4 is 2 the square root of 9 is 3

and the square root of 49 is 7.

so 4 times 2 is 8.

and then 8 times 3 is 7 I mean no 8

times 3 is 24. but 24 times 7 that's

going to give us the same answer of 168

so we know we have the right answer

so that's it for this video now you know

two methods that you can employ in order

to calculate the area of an isosceles

triangle