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Here is a diagram of a black circle inside a white square and a diagram of a small grey square inside a white square.
The two white squares are the same size.


AB = 8x.

The sides of the square are tangents to the circle at the points E, F, G and H.
W, X, Y and Z are the midpoints of AB, BC, CD and DA respectively.

Work out the ratio of the area of the black circle to the area of the white square to the area of the grey square. Give your answer in its simplest form.

Hi, I don't understand this. Can someone please explain? It's 50 points.

Here is a diagram of a black circle inside a white square and a diagram of a small grey square inside a white square The two white squares are the same size AB class=

Respuesta :

MrRoyal

The ratio of the area of the black circle to the area of the white square to the area of the gray square is π : 4 : 2

How to determine the ratio of the areas?

The side length AB is given as:

AB = 8x

The area of the white square is calculated as

White = (AB)²

This gives

White = (8x)²

Evaluate

White = 64x²

The side length AB represents the diameter of the black circle.

So, the radius is:

r =AB/2

This gives

r = 8x/2 = 4x

The area is then calculated as:

Circle = πr²

This gives

Circle = π(4x)²

Evaluate

Circle = 16πx²

Next, calculate AW and AZ using:

AZ = AW = AB/2

Evaluate

AZ = AW = 8x/2 = 4x

Calculate WZ using the following Pythagoras theorem

WZ² = AW² + AZ²

This gives

WZ² = (4x)² + (4x)²

Evaluate

WZ² = 2(4x)²

Take the square root of both sides

WZ = 4x√2

The area of the gray square is:

Gray = WZ²

This gives

Gray = 2(4x)²

Evaluate

Gray = 32x²

At this point, the areas are:

  • White = 64x²
  • Circle = 16πx²
  • Gray = 32x²

The ratio is then represented as:

Ratio = Circle : White : Gray

This gives

Ratio = 16πx² : 64x² : 32x²

Divide through by 16x²

Ratio = π : 4 : 2

Hence, the ratio of the area of the black circle to the area of the white square to the area of the gray square is π : 4 : 2

Read more about areas at:

https://brainly.com/question/24487155

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