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1. A given four-sided figure has one of its angles a right angle; construct a right-angled triangle equal to it in area.
2. On a given finite straight line as diagonal construct a rhombus equal to a given parallelogram.
3. To a given straight line apply a rectangle equal in area to a given rectilineal figure of four sides.
151. To describe a parallelogram equal in area to a given rectilineal figure, having one side equal to a given finite straight line, and one angle equal to a given angle.
Let A be the given rectilineal figure,
let BC be the given finite straight line, D the given angle ; it is required to describe a parallelogram equal in area to A, having one side equal to BC and one angle equal to D.
Describe the triangle E equal in area to A; [Prop. C]+ then describe the parallelogram BCFG
equal in area to E, having BC for one of its sides,
and having the angle BCF equal to D; [Prop. 44] then BCFG is the parallelogram required.
Proof. Because the triangle E is equal in area to A, therefore the parallelogram BCFG is equal in area to A, and BC is one of its sides and the angle CBF is equal to D. Wherefore, a parallelogram has been described equal in area to the given figure, etc.
When this proposition is set in a Euclid examination, the student would be expected to write out the construction and proof of Prop. C, instead of merely quoting it.
1. To a given straight line apply a right-angled triangle equal in area to a given rectilineal figure of five sides.
Prove that the area of a trapezium is equal to that of a parallelogram between the same parallels whose base is equal to half the sum of the unequal sides of the trapezium.
[Def. A Trapezium is a four-sided figure having a pair of parallel sides.]
3. Describe a rectangle equal in area to a given trapezium.
4. Shew how to solve Proposition 45, without the aid of Prop. C, by dividing the rectilineal figure up into triangles and making repeated use of Prop. 44. (This is the method of proof given by Euclid.)
152. To describe a square upon a given straight line.
Let AB be the given straight line;
it is required to describe a square upon AB.
Construction. At the point A in the straight line AB draw AC at right angles to AB;
[Prop. 11] in AC take D so that AD is equal to AB; [Prop. 3]
through B draw BE parallel to AD;
then ABED is the square required.
[Prop. 31] [Prop. 31]
Proof. Because ABED is a parallelogram
therefore it is a square, [Def. Art. 130] and it is described upon the given line AB. Wherefore, a square has been described, etc.
1. Prove that a square has all its sides equal and all its angles right angles.
2. Prove that the squares described upon equal straight lines are equal in all respects.
3. Prove that if two squares are of equal area they are described upon equal straight lines.
4. Prove that the diagonal of a square makes an angle of 45° with the sides.
5. The diagonals of a square bisect each other at right angles.
6. If the diagonals of a parallelogram are equal to and at right angles to each other the parallelogram is a square and not otherwise.
7. Construct a square equal in area to the sum (i) of three given squares, (ii) of four given squares.
8. Construct a square whose area is equal (i) to twice, (ii) to thrice, (iii) to four times that of a given square.
9. Describe a square whose area is equal to the difference of two given squares.
10. Construct a square such that two of its opposite sides pass through two given points and having a portion of a given straight line as a third side.
CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS.