A rectangle has a width of 4 cm and a height of 3 cm. If another rectangle is to be drawn so it has a golden ratio with the first rectangle, what could the dimensions of the new rectangle be?
7 cm by 4 cm
8 cm by 3 cm
3 cm by 4 cm
4 cm by 8 cm
To find a rectangle with a golden ratio to the first rectangle, we need to understand what the golden ratio is. The golden ratio, denoted by the Greek letter phi (φ), is approximately 1.618.
The golden ratio states that if we divide a line into two parts such that the ratio between the whole line and the longer segment is the same as the ratio between the longer segment and the shorter segment, then the ratio is equal to the golden ratio.
In terms of rectangles, the golden ratio occurs when the ratio of the longer side to the shorter side is equal to approximately 1.618.
Let's find the longer side of the first rectangle using the given dimensions:
- Width = 4 cm
- Height = 3 cm
The longer side of the rectangle can be either the width or the height, so let's compare both ratios:
- Width ratio: 4 cm / 3 cm ≈ 1.333
- Height ratio: 3 cm / 4 cm = 0.75
Since the width ratio is greater than the height ratio, we need to find a rectangle with a width-to-height ratio that is approximately 1.618.
Now, let's evaluate the dimensions provided for the new rectangle:
- 7 cm by 4 cm: Width ratio = 7 cm / 4 cm = 1.75 (not close to the golden ratio)
- 8 cm by 3 cm: Width ratio = 8 cm / 3 cm ≈ 2.667 (not close to the golden ratio)
- 3 cm by 4 cm: Width ratio = 3 cm / 4 cm = 0.75 (close to the golden ratio!)
- 4 cm by 8 cm: Width ratio = 4 cm / 8 cm = 0.5 (not close to the golden ratio)
From the given options, the rectangle with dimensions of 3 cm by 4 cm has a width-to-height ratio that is closest to the golden ratio of approximately 1.618.