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gtnhenbr [62]
3 years ago
15

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
Mathematics
1 answer:
Lera25 [3.4K]3 years ago
6 0

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

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