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# M5.4.10

Constructs the image of a geometric figure under a given transformation (e.g., subdividing, dilating) and determines the relationship of the areas and perimeters of the two figures.

### Isoperimetric Geometry

The isoperimetric theorem states that: "Among all shapes with an equal area, the circle will be characterized by the smallest perimeter" which is equivalent to "Among all shapes with equal perimeter, the circle will be characterized by the largest area." The theorem's name derives from three Greek words: 'isos' meaning 'same', 'peri' meaning 'around' and 'metron' meaning 'measure'. A perimeter (= 'peri' + 'metron') is the arc length along the boundary of a closed two-dimensional region (= a planar shape).

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### Can an Astronaut on Mars distinguish the Earth from its Moon?

Some day an astronaut will stand on Mars and look back at Earth. As Alfred, Lord Tennyson wrote Venus, Hesper, Were we native to that splendour or in Mars We would see the globe we groan in, fairest of their evening stars Could we dream of wars and carnage, craft and madness lust and spite Roaring London, raving Paris, in that peaceful point of light? Would we not, when gazing heavenward, at a star so silver-fair Yearn, and clasp the hands, and murmur: Would to God that we were there? But what exactly will one see from Mars?

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