Question

A right triangle has two legs with lengths a and b and a hypotenuse with leg c. In this triangle, the area and perimeter are the same nonzero number. Find the length of a if b=6. (Hint: use the Pythagorean Theorum and solve algebraically).

Answer 1

of you know the area of a triangle and the perimeter
write those formulas given you know b=6

Answer 2

we will also need the Pythagorean theorem

Answer 3

what you need:
A=.5ab
P=a+b+c
a^2+b^2=c^2

you are given b=6
You are also given A=P so let's rewrite those:
P=.5a(6)=3a
P=a+(6)+c
Use these two equations to write an equation in terms of two variables.
Then use that equation and sub into
a^2+6^2=c^2

Answer 4

let me know if you still don't understand

Answer 5

hi myininaya, i think i understand, but i'm confused on how to combine the two equations. c introduces another variable which i believe complicates things. thank you so much for your help!! :)

Answer 6

So A and P are equal that is why we can set this equation
3a=a+c+6
2a=c+6
so c=2a-6

now go to a^2+6^2=c^2
and sub the c in this equation
and solve for a

Answer 7

you will only have one variable

Answer 8

So you will find
a then you go back to c=2a-6 to find c

Answer 9

so i plug in 2a-6 in for c in the pythagorean theorum?

Answer 10

:) yep

Answer 11

thank you so much! oh wait whad do i plug in for a ????

Answer 12

you are solving that equation for a

Answer 13

and does it plug in for c or c^2?

Answer 14

nothing to plug in for a
you are trying to come up with an equation that contains one variable
for c
just replace the c part
make sure you keep that exponent 2
because we haven't done anything with it yet

Answer 15

do i also square the six?

Answer 16

we have a^2+6^2=c^2
replacing c with 2a-6 with
a^2+6^2=(2a-6)^2


You need to multiply (2a-6)(2a-6)

Answer 17

ah! thank you bunches, if i could give you a million medals i would, i think i finally understand!!!!!!

Answer 18

good news :)

Answer 19

so i got it to a=4a^2-24a. i was never very good at algebra, do i put the 24a on the other side and then find the square root of both sides, and then divide both sides by the square root of the right side?

Answer 20

is that a^2=4a^2-24a?

Answer 21

yes. i just do not know what to do with it from here

Answer 22

Right it as a degree 2 polynomial =0

Answer 23

Or get everything on one side is what i'm saying

Answer 24

so now i have 25a-4a^2=0???

Answer 25

I thought you had a^2=4a^2-24a?

Answer 26

yes i did, but then i moved everything to one side

Answer 27

so a^2-4a^2+24a=0

Answer 28

a^2 and 24a aren't liek terms

Answer 29

OH I FORGOT TO SQUARE THE A

Answer 30

a^2 and -4a^2 are

Answer 31

but that would give you -3a -4a^2

Answer 32

no wouldn't...

Answer 33

what is a^2-4a^2?

Answer 34

oh it would give you -3a^2+24a

Answer 35

yes -3a^2+24a=0

Answer 36

now both of the terms have -3a in common so factor -3a out

Answer 37

i got a^2-8a

Answer 38

do i leave it like that?

Answer 39

OH IS A=8!

Answer 40

so so if you factored out -3a
why do you have a^2 there?

Answer 41

and yes a=8

Answer 42

-3a^2+24a=0
3a(a-8)=0
a-8=0
a=8

Answer 43

thank you you are brilliant :)))))))))

Answer 44

because obviously one of the lengths can't be the other answer which is a=0

Answer 45

it wouldn't be a triangle lol

Answer 46

i did it differently but still got the same answer?

Answer 47

i divided both sides by just -3, and got a^2-8a=0, and then i moved the 8a to the other side to make it positive, and then divided both sides by a

Answer 48

well just so you know usually you can't divide both sides by a because a can be zero
but a actually cannot be 0 here so that is fine
just put next to that step a not equal to 0