How many ways can the letters of the word algebra be arranged?
1 answer:
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Answer:
Shorter leg: 8 units
Longer leg: 15 units
Hypotenuse: 17 units
Step-by-step explanation:
We are given where
is the hypotenuse and
is the length of the shorter leg.
We got this equation from reading that the "hypotenuse of a triangle is 1 foot more than twice the length of the shorter leg". I replaced the "hypotenuse of a triangle" with , "is" with
, "1 foot more than" with
and finally "twice the length of the shorter leg" with
.
We also have "longer leg is 7 feet longer than the shorter leg".
I'm going to replace "longer leg" with .
I'm going to replace "is" with .
I'm going to replace "7 feet longer than the shorter leg" with .
So we have the equation .
So we have a right triangle since something there is a side being referred to as the hypotenuse. We can use Pythagorean Theorem to find a relation between all these sides.
So by Pythagorean Theorem, we have: .
Let's make some substitutions from above:
Let's expand the powers using:
Applying this now:
Combine like terms on right hand side:
Subtract everything on left hand side to get 0 on that side:
Simplify:
Reorder into standard form for a quadratic:
Every term is even and therefore divisible by 2. I will divide both sides by 2:
I'm going to see if this is factoroable.
We need to see if we can come up with two numbers that multiply -24 and add up to be -5.
Those numbers are -8 and 3.
So the factored form is:
This implies that either os
.
The first equation can be solved by adding 8 on both sides: .
The second equation can be solved by subtracting 3 on both sides: .
The only solution that makes sense for is 8 since it can't the shorter length cannot be a negative number.
So the dimensions of the right triangle are:
Shorter leg: 8 units
Longer leg: 15 units
Hypotenuse: 17 units