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

Answer:

Shorter leg: 8 units

Longer leg: 15 units

Hypotenuse: 17 units

Step-by-step 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