Question

Do you hypotenuse of a triangle is 1 foot more than twice the length of the shorter leg the longer leg is 7 feet longer than the shorter leg find the dimensions of the triangle

Answer 1

Shorter leg: 8 units

Longer leg: 15 units

Hypotenuse: 17 units

Explanation:

We are given
`h=1+2s` where
`h` is the hypotenuse and
`s` 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
`h`, "is" with
`=`, "1 foot more than" with
`1+` and finally "twice the length of the shorter leg" with
`2s`.

We also have "longer leg is 7 feet longer than the shorter leg".

I'm going to replace "longer leg" with
`L`.

I'm going to replace "is" with
`=`.

I'm going to replace "7 feet longer than the shorter leg" with
`7+s`.

So we have the equation
`L=7+s`.

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:
`s^2+L^2=h^2`.

Let's make some substitutions from above:

`s^2+(7+s)^2=(1+2s)^2`

Let's expand the powers using:

`(a+b)^2=a^2+2ab+b^2`

Applying this now:

`s^2+(49+2(7)s+s^2)=(1+2(1)(2s)+(2s)^2)`

`s^2+49+14s+s^2=1+4s+4s^2`

Combine like terms on right hand side:

`2s^2+49+14s=1+4s+4s^2`

Subtract everything on left hand side to get 0 on that side:

`0=(1-49)+(4s-14s)+(4s^2-2s^2)`

Simplify:

`0=(-48)+(-10s)+(2s^2)`

Reorder into standard form for a quadratic:

`0=2s^2-10s-48`

Every term is even and therefore divisible by 2. I will divide both sides by 2:

`0=s^2-5s-24`

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:

`0=(s-8)(s+3)`

This implies that either
`s-8=0` os
`s+3=0`.

The first equation can be solved by adding 8 on both sides:
`s=8`.

The second equation can be solved by subtracting 3 on both sides:
`s=-3`.

The only solution that makes sense for
`s` is 8 since it can't the shorter length cannot be a negative number.

`s=8`

`L=7+s=7+8=15`

`h=1+2s=1+2(8)=1+16=17`

So the dimensions of the right triangle are:

Shorter leg: 8 units

Longer leg: 15 units

Hypotenuse: 17 units