PROP. 11. PROB.

To divide a given straight line into two parts so that the square on one part shall be equal to the rectangle contained by the whole and the other part.

Let AB be the given straight line.

To divide AB as at P, so that AB. BP

Describe the sq. on AB (I. 46), bisect AC at E (I. 10).

Join EB and produce EA, making EF = EB.

On AF describe a square (I. 46).

P is the point required. Let GP meet CD at K.
CE and EF are two lines,

.. CF. FA= the difference of EF and EA2 (Prop. A),
- the difference of EB2 and EA2,

=

= AB2.

But CF. FA is FK, and AB is AD,

:: FK

= AD. Take away the common part AK.

But FP is AP2, and PD is rect. DB.BP

.. AP2 = AB. BP. Q.E.F.

Note 1.-In the second figure P is in AB produced. The only modifications in the above required for this figure are, for "produce EA," read "produce · AE;" and in the proof, instead of “take away," read "add."

Note 2.-Proposition 11 solves very nicely a problem which in algebra is solved by means of a quadratic equation, and thus furnishes an interesting geometrical illustration of the double solution of the quadratic equation. Let a be the given line, and let x be the part to be squared; then x2a (ax) is the equation, which, on being solved, gives

2

These two values of x correspond respectively to AP in Fig. 1 and AP in Fig. 2.

Let A be the given figure.

To make a square

=

A. |

Make a rectangular BCDE = A (I. 45).
If BC= CD, then BD is the sq. required.

If not, produce BC, making CF = CD.

Bisect BF at G, and describe a semicircle with G as

centre and GB as radius.

Produce DC to meet the circumference at H.

HC is a side of the sq. required. Join GH.

BG is one line, and GC another,

.. BC. CF = the difference of BG2 and GC2 (Prop. A),

=

=

But BC.CF

the difference of GH2 and GC2,

HC2.

=

.. CH2

18. A straight line is divided into two parts; if twice the rectangle contained by the parts be equal to the sum of the squares on the two parts, the line shall be bisected.

19. In a right-angled triangle the square on either of the sides containing the right angle equals the rectangle contained by the sum and difference of the other two sides.

20. In the hypotenuse BC of a right-angled triangle ABC, two points D and E are taken such that BD = BA and CE CA; show that DE2 2 rect. BE. CD.

=

21. Construct a rectangle equal to a given square, and so that the difference of its sides equals a given line.

22. Two rectangles have equal areas and perimeters; show that they are congruent.

23. Divide a straight line into two parts such that the square on one part may be equal to half the square on the

whole line.

24. Divide a straight line into two parts such that the rectangle contained by them shall be equal to the square on a given line. What is the limit to the size of the latter given line?

25. Four times the sum of the squares on the three medians of a triangle equals three times the sum of the squares on the sides of the triangle.

26. Divide a given straight line into two parts so that the square on one part may be double of the square on the other.

27. Show that the square on a straight line drawn from the right angle of a right-angled triangle perpendicular to the hypotenuse is equal to the rectangle contained by the segments of the hypotenuse; also that the square on a side of the right-angled triangle equals the rectangle contained by the hypotenuse and the segment of the hypotenuse adjacent to that side.

28. Describe an isosceles triangle such that the square on the base may be equal to three times the square on one of the equal sides.

29. Show that the difference of the squares on two sides of a triangle equals the rectangle contained by the base and the difference of the segments into which it is divided by the perpendicular upon it from the vertex.

30. Hence, if the three sides of a triangle be given (say 13, 14, and 15), find the segments of the base made by the perpendicular upon it from the vertex, and also the length of this perpendicular.

Miscellaneous Exercises on Books I. and II.

=

1. In the figure of I. 47, prove KD2 + FE2 5 BC2; DA2 + AC2 = ÉA2 + AB2; also that BF, CK, and AL are concurrent.

2. Show that in a straight line divided as in II. 11 the rectangle contained by the sum and difference of the parts is equal to the rectangle contained by the parts.

3. The triangle formed by joining the middle points of the sides of a triangle is a fourth of the whole triangle.

4. The lines joining the middle points of the sides of a quadrilateral form a parallelogram equal in area to half the quadrilateral.

5. In a quadrilateral the squares on the diagonals together equal twice the sum of the squares on the lines joining the middle points of the opposite sides.

6. If the middle points of two opposite sides of a quadrilateral be each joined to the middle points of the two diagonals, the figure formed will be a parallelogram.

7. If the middle points of the opposite sides of a quadrilateral be joined, they will intersect on the line joining the middle points of the two diagonals, and will be bisected there.

8. The middle points of the three diagonals of a complete quadrilateral are collinear (i.e., in the same straight line).

Note. The third diagonal of a quadrilateral is the line joining the points of intersection of the pairs of opposite sides.

9. In AB, the diameter of a circle, take two points C and D equally distant from the centre, and join C and D to E, any point in the circumference; then EC2+ED2 = AC2 + AD2.

10. The square on any straight line drawn from the vertex of an isosceles triangle to the base is less than the square on one of the equal sides by the rectangle contained by the segments of the base.

11. Find the locus of a point, the sum or the difference of its distances from two fixed lines being constant.

12. Produce a given line so that the rectangle contained by the whole line thus produced and the part produced shall be equal to the square on the given line.

13. If the square on the perpendicular from the vertex of a triangle to the base be equal to the rectangle contained by the_segments of the base, the vertical angle will be a right angle.

14. If a straight line be divided into two unequal parts, show that the squares on the two parts are together equal to twice the rectangle contained by the parts, and four times the square on the line between the point of section and the middle of the line.

15. From a given point to draw to two parallel straight lines two straight lines and one another.

16. The area of a rhombus is equal to half the area of the rectangle contained by its two diagonals.

17. In AB, the hypotenuse of a right-angled triangle ABC, take any point D and draw DE LAB, meeting AC at E; show that rect. AB. AD rect. AC. AE.

=

18. In triangle ABC, the angles B and C are acute, and lines are drawn from B and C perpendicular to the opposite sides, meeting them in E and F; show that the square on BC equals the sum of the rectangles AB. BF, AC. CÊ.

19. If x and y represent any integers, show that lines whose lengths are x2 + y2, 2 xy, and x2 - y2, could form a rightangled triangle.

20. In the figure to I. 18, prove angle DBC equal to half the difference of the angles ABC and ACB.

21. The sum of the perpendiculars to the sides of an equilateral triangle from any point within it will be equal to the perpendicular from one of its angles to the opposite side. What change will be required in the enunciation if the point be taken outside the triangle?

22. The square on the hypotenuse of a right-angled isosceles triangle is four times the area of the triangle.

23. In the figure of II. 11, prove CP produced if CP and BE meet at O, AO 1 CP.

BF; also

24. If ABC be a triangle right-angled at C, and AD, BE be drawn bisecting the opposite sides, then four times the sum of the squares on AD and BE shall be equal to five times the square on AB.

25. If any point be taken in the plane of a parallelogram, and perpendiculars be drawn from it to two adjacent sides of the parallelogram and the diagonal between them, the rectangle contained by the diagonal and the perpendicular upon it shall be equal to the sum or the difference of the rectangles contained by the two sides and the perpendiculars on them.

EDINBURGH: PRINTED BY OLIVER AND BOYD.