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Question

the width of a rectangular gate is 2 meters (m) larger than its height.

The diagonal brace measures (square root of 6) meters. Find the width and height.

Answer 1

w = h + 2

d ² = w ² + h ²

6 ² = (h + 2) ² + h ²

36 = h ² + 4h + 4 + h ²

2 h ² + 4 h + 4 = 36

h ² + 2 h + 2 = 18

h ² + 2 h - 16 = 0

h = [ - 2 ± √ (4 + 64 ) ] / 2

h = [ - 2 ± √ (68) ] / 2

h = 3.12 m

w = 5.12 m

Answer 2

The diagonal brace divides the rectangular gate into 2 (identic) rectangular triangles. So this can be solved by the Pythagorean theorem

a² + b² = c²

http://en.wikipedia.org/wiki/Pythagorean...

a is height,  b is width in that case, c is the diagonal brace.

So you come to the following equation:

a² + b² = (sqrt 6)²

a² + b² = 6

And you know:

b = a + 2

Replace b in formula above:

a² + ( a + 2 )² = 6

a² + a² + 4a + 4 = 6

2a² + 4 a - 2 = 0

Solve with quadratic formula:

http://en.wikipedia.org/wiki/Quadratic_e...

You now have the value for a.  And as as b = a + 2 finding b will be easy.. ;o)

Answer 3

Let

H = height of rectangulat gate

W = width of rectabgular gate

D = diagonal brace

H = ?

W = H + 2

D = sqrt 6  (sqrt = square root)

If you draw the figure, you will create 2 right triangles.

If you have a right triangle, the pythagorean theorem or formula is the handy solution.

D becomes the hypothenuse

W is the base

H is the height or the other side

D^2 = W^2 + H^2

Since W = ( H + 2)

and D = sqrt 6

(sqrt 6)^2 = (H + 2)^2 + H^2

6 =( H^2 + 4H + 4) + H^2

6 = 2H^2 + 4H + 4

Divide by 2 to simply the equation

3 = H^2 + 2H + 2

3 - 3 = H^2 + 2H + 2 - 3

0 = H^2 + 2H - 1

H^2 + 2H - 1 = 0  This is not a perfect square. We can use the quadratic formula to solve for H.

a = 1, b = 2 , c = -1

H = (-b +/- sqrt b^2 - 4ac)/ 2a

H = ( - 2 +/- sqrt 2^2 - 4(1)(-1)]/ 2(1)

H = (-2 +/- sqrt 4 + 4)/2

H = (-2 +/- sqrt 8)/2

H = (-2 +/- sqrt 2^2 * 2)/2

H = (-2 +/- 2 sqr t2)/2

H =( -2 + 2 sqrt 2)/2 =  -2/2 + 2 sqrt2/2 = - 1 +  sqrt 2 = -1 + 1.414 =

H = 0.414 m

H = (-2  - 2 sqrt 2)/2 = -2/2 - 2sqrt 2/2 = -1 - sqrt 2 = -1 - 1.414 =

H = -2.414 m ==> disregard this value

The height of the gate is 0.414 meter