Question 95181
Without going into all the detail, here's a way you can do it.
.
First let x be one leg of the triangle. Then the hypotenuse is x + 7. Call the other leg
of the triangle y.
.
This means that the perimeter of the triangle (which is given as 392 ft) is the sum of the
three sides. In equation form this is:
.
{{{y + x + x+7 = 392}}}
.
Combine the x terms on the left side to get
.
{{{y + 2x + 7 = 392}}}
.
Solve for y by subtracting 2x + 7 from both sides to get:
.
{{{y = 392 - 2x - 7 = 385 - 2x}}}
.
Now let's switch from the perimeter to the Pythagorean theorem. The sum of the squares of
the two legs equals the square of the hypotenuse. In equation form, for this problem the
Pythagorean theorem can be expressed in equation form as:
.
{{{y^2 + x^2 = (x+7)^2}}}
.
Square out the right side and you get:
.
{{{y^2 + x^2 = x^2 + 14x + 49}}}
.
Subtract {{{x^2}}} from both sides and you have:
.
{{{y^2 = 14x + 49}}}
.
Solve for y by taking the square root of both sides:
.
{{{y = sqrt(14x + 49)}}}
.
So now we have two equations for y:
.
{{{y = 385 - 2x}}} and
{{{y = sqrt(14x + 49)}}}
.
Since the left sides of these two equations are equal (both left sides are y), the right
sides must also be equal. So set the right sides equal:
.
{{{385 - 2x = sqrt(14x + 49)}}}
.
Get rid of the radical by squaring both sides to get:
.
{{{148225 - 1540x + 4x^2 = 14x + 49}}}
.
Subtract 14x + 49 from both sides:
.
{{{148176 - 1554x + 4x^2 = 0}}}
.
Rearrange terms in descending powers of x:
.
{{{4x^2 - 1554x + 148176 = 0}}}
.
This is now in the standard form of a quadratic equation. Use the quadratic formula to
solve it and you will get two answers for x. The two answers are:
.
{{{x = 441/2 = 220.5}}} and
{{{x = 168}}}
.
The first answer gets tossed out because if x = 220.5 then the hypotenuse is 227.5 and
adding just these two gives an answer that is bigger than the perimeter.
.
So the acceptable answer is (as you already knew) x = 168. This makes the hypotenuse
175, and therefore, y = 49.
.
Hope this helps.