Question

A rectangular piece of wood 4cm by 12cm tall is tilted at an angle of 45degre. Find the vertical distance between the lower corner and the upper corner.

Answer 1

• The lower corner is the corner touching the ground.

• The upper corner is the one furthest from the floor.

We can determine that from the lower 2 corners to where the floor and wall intersect are 2√2 cm long.

Apply Pythagorean theorem,

$\bold{4^2=x^2+x^2}$

$\implies\bold{16=2x^2}$

$\implies\bold{x^2=\frac{16}{2}}$

$\implies\bold{x^2=8}$

$\boxed{\therefore{\bold{x=2\sqrt2}}}$

We know that the two sides are equal since the angles in the triangle are 90:45:45

We need to know the remainder of the vertical distance you want after taking away 2√2.

So we want to make a new triangle with either of the sides that equal 12.

Took one of them touches the floor and one endpoint.

We know that this is another 90:45:45 triangle.

So we can do the Pythagorean theorem again and set the a and b equal.

$\bold{12^2=x^2+x^2}$

$\implies\bold{144=2x^2}$

$\implies\bold{x^2=72}$

$\boxed{\bold{\therefore x=6\sqrt2}}$

$\bold{\: Total\: vertical\: distance=2\sqrt2+6\sqrt2}$

$\boxed{\therefore\bold{\: Total\: Vertical\: Distance=8\sqrt2}}$

Answer 2

Given,

Dimensions of a rectangular piece of wood:

Length = L = 12 centimeters = 12 cm

Breadth = B = 4 cm

The angle of inclination of the wood piece = 45 degrees

To find,

The vertical distance between the lower corner and the upper corner of the rectangular piece of wood.

Solution,

We can simply solve this mathematical problem using the following process:

As the rectangular piece of wood is inclined at 45 degrees, hence, the vertical and horizontal distance between the lower corner and the upper corner of the rectangular piece of wood are both equal and this inclination forms an isosceles right-angled triangle with the length of the wood piece forming its hypotenuse.

Let us assume that the vertical distance between the lower corner and the upper corner of the rectangular piece of wood = the corresponding horizontal distance = x centimeters

Now, on applying the Pythagoras theorem for the isosceles right-angled triangle, we get:

(vertical distance)^2 + (horizontal distance)^2 = (hypotenuse)^2 = (Lenght of the wood piece )^2

=> x^2 + x^2 = (12)^2 = 144

=> x^2 = 72

=> x = √72

=> x = 6√2 cm

Hence, the vertical distance between the lower corner and the upper corner of the rectangular piece of wood is 6√2 centimeters.