Question

How do you solve this?

Mr. Hamilton is placing a support plank along the diagonal of a gate. The height of the gate is 5 feet, and the diagonal is 1 foot longer than the width of the gate, as shown below.

What is the width, in feet, of the gate?

Answer 1

`12` feet

Explanation:

By Pythagoras:

`(w+1)^2 = w^2+5^2`

That is:

`w^2+2w+1 = w^2+25`

Subtracting `w^2+1` from both sides, we find:

`2w = 24`

Divide both sides by `2` to get:

`w = 12`

So the gate is `12` feet wide.

`color(white)()`
Footnote

The first few right angled triangles with sides `a`, `b` and `b+1` have sides of lengths:

`3, 4, 5`

`5, 12, 13`

`7, 24, 25`

`9, 40, 41`

In general, they take the form:

`a, (a^2-1)/2, (a^2+1)/2`

for any odd value of `a >= 3`

Answer 2

`"width "=12" feet"`

Explanation:

Using `color(blue)"Pythagoras' theorem"` on the right triangle.

`color(orange)"Reminder"`

The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides.

`rArr(w+1)^2=w^2+5^2`

`rArrw^2+2w+1=w^2+25`

subtract `w^2` from both sides.

`cancel(w^2)cancel(-w^2)+2w+1=cancel(w^2)cancel(-w^2)+25`

`rArr2w+1=25`

subtract 1 from both sides.

`2wcancel(+1)cancel(-1)=25-1`

`rArr2w=24`

divide both sides by 2

`(cancel(2) w)/cancel(2)=24/2`

`rArrw=12" is the solution"`

That is, the width of the gate is 12 feet.