EXTENSIONS OF PYTHAGOras' TheoreM.

THEOREM 31. In an obtuse-angled triangle, the square on the side opposite to the obtuse angle is equal to the sum of the squares on the sides containing the obtuse angle plus twice the rectangle contained by one of those sides and the projection* on it of the other.

Data

Let BC

The AABC has BAC obtuse.

CN is the perpendicular from C upon BA (produced),

.. AN is the projection of AC upon BA.

= a units, CA = b units, AB = c units, AN = p units, CN= h units.

* The projection of a finite straight line upon another straight line is the length intercepted on the second line between the perpendiculars drawn to it from the extremities of the former,

S. H. T. G.

5

THEOREM 32. In any triangle, the square on the side opposite to an acute angle is equal to the sum of the squares on the sides containing that acute angle minus twice the rectangle contained by one of those sides and the projection on it of the other.

CN is the perpendicular from C upon AB (or AB produced),

.. AN is the projection of AC upon AB.

Let BC = a units, CA = 6 units, AB = c units, AN = p units, CN = h units. BC2 = CA2 + AB2 - 2AB. AN,

To prove that

Proof

Since ▲ BNC is right-angled, .. BC2 = BN2 + NC2,
i.e. in fig. 67, a2 = (c-p)2 + h2,

in fig. 68, a2 = (p −c)2 + h2,

.. in both figures, a2 = c2 - 2cp + p2 + h2.

But

ANC is right-angled, .. p2 + h2 = b2,
• . a = c − 2cp+b*,

i.e. BC2 AB2 - 2AB. AN + AC2.

=

Q. E. D.

Pyth.

Pyth.

NOTE ON THEOREMS 31, 32. Euclid makes two theorems of Theorems 31 and 32, but from a modern standpoint they are two cases of one theorem, which may be expressed trigonometrically.

In figs. 67, 68, p = b cos A; and a2 = b2 + c2 — 2bc cos A.

Pythagoras' theorem is a particular case of this; for if A= 90°, cos A = 0.

APOLLONIUS' THEOREM*.

THEOREM 33. In any triangle, the sum of the squares on any two sides is equal to twice the square on half the third side together with twice the square on the median which bisects the third side.

Data ABC is a triangle, D is the mid-pt.

Ex. 8. Show geometrically that (i) √a2+b2 is not a+b; (ii) √a2 – b2 is not

a-b.

Ex. 9. Show how to construct a square whose area is twice that of a given square.

Ex. 10. Given two squares of different sizes, show how to construct a square equal to (i) their sum, (ii) their difference.

Ex. 11. AD is the altitude of a triangle ABC. Prove that

AB2-AC2BD2-CD2.

Ex. 12. A point moves so that the difference of the squares of its distances from two given points is equal to a given square. Prove that the locus of the point is a straight line.

Ex. 13. PQR is a triangle, right-angled at Q. On QR a point S is taken. Prove that PS2+QR2= PR2+QS2.

Ex. 14. The diagonals of a quadrilateral ABCD intersect at right angles. Show that AB2+ CD2=BC2+ DA2.

* Apollonius was a Greek who studied at Alexandria: he lived about half a century after Euclid and Archimedes. His nickname was Epsilon.

Ex. 15. ABC is a triangle, right-angled at A. On AB, AC respectively points X, Y are taken. Prove that BY2+CX2=XY2+ BC2.

Ex. 16. ABC is a triangle right-angled at B, D is the mid-point of BC. Prove that AD2=AC2-3 BD2.

Ex. 17. O is a point inside a rectangle ABCD. Prove that

OA2+ OC2=OB2+ OD2.

Ex. 18. The sum of the squares on the sides of a rhombus is equal to the sum of the squares on its diagonals.

Ex. 19. In an equilateral triangle the sum of the squares on the three sides is equal to four times the square on the perpendicular from a vertex to the base. Ex. 20. A straight line AB is produced to C, so that AC=3AB; on BC an equilateral triangle BCD is described. Prove that the square on AD is seven times the square on AB.

Ex. 21. BDEC is the square described on the hypotenuse BC of the rightangled triangle ABC, and AF is the perpendicular from A to BC; if BC=a, CA=b, AB=c, AF=p, find an expression for AD2 in terms of a, b, c, p; and show that AD2 + AC2= AE2 + AB2.

Ex. 22. ABC is any triangle and from any point P within it perpendiculars PX, PY, PZ are drawn to BC, CA, AB respectively. Prove that

BX2+CY2+AZ2=CX2+BZ2+AY2.

Ex. 23. The perpendiculars drawn from B, C to the sides AC, AB of the triangle ABC meet at P; prove that AC2+BP2=AB2+CP2.

[You may assume that the perpendicular from A to BC passes through P.] Ex. 24. ABC is an isosceles triangle right-angled at C, and D is any point on AB; prove that AD2+ DB2=2CD2.

Ex. 25. Prove that the square on the difference of the sides of a right-angled triangle, together with twice the rectangle contained by the sides, is equal to the square on the hypotenuse. (Use Algebra.)

Ex. 26. Half the area of a square is cut away in the form of a square about a diagonal, one corner being common to the two squares. Prove that the remaining figure is divided by the diagonals of the original square into four parts which may be re-united into a square.

Ex. 27. On the diagonal AB of a square two points P and Q are taken such that AQ, PB are each equal to a side of the square; prove that the square on PQ is twice that on AP.

EXTENSIONS OF PYTHAGORAS' THEOREM.

Ex. 28. Prove that the projections on the same straight line of equal and parallel straight lines are equal.

Ex. 29. Prove that, if the slope of a line is 60°, its projection is equal to half the line.

AC=8 cm.,

Ex. 30. Calculate the side BC of a ▲ ABC when (i) AB=10 cm., ▲ A=60°, (ii) AB=10 cm., AC=8 cm., ▲ A=120°.

Ex. 31. Calculate the area of the rectangle referred to in the enunciation of Ths. 31, 32 for the following cases:3:—(i) c=3 in., b=2 in., a = 4 in., (ii) c=3 in., b=2 in., a=2 in.

Ex. 32. By comparing the square on one side with the sum of the squares on the other two sides, determine whether triangles having the following sides are acute-, obtuse-, or right-angled:

(i) 3, 4, 6; (ii) 3, 4, 3; (iii) 2, 3, 5; (iv) 2, 3, 4; (v) 12, 13, 5.

Ex. 33. Given four sticks of lengths 2, 3, 4, 5 feet, how many triangles can be made by using three sticks at a time? Find out whether each triangle is acute-, obtuse-, or right-angled.

TEx. 34. What does Th. 32 become if B is a right angle?

TEx. 35. Suppose that A in Fig. 66 becomes larger and larger till BAC is a straight line. What does Th. 31 become in this case?

¶Ex. 36. Suppose that ▲ A in Fig. 67 becomes smaller and smaller till C is on BA. What does Th. 32 become in this case?

Ex. 37. In the triangle ABC, BAC is an obtuse angle, BD and CE are drawn at right angles to CA, BA respectively. Prove the rectangle BA. AE equal to the rectangle CA. AD.

Ex. 38. ABC is a triangle in which AB is 7 in., BC is 5 in., CA is 3 in. The circle whose centre is A and radius is AC cuts BC again in D. Prove that ACD is an equilateral triangle.

Ex. 39. In the trapezium ABCD (fig. 70), prove that AC+BD2 AD2+ BC2+2AB.CD.

Ex. 40. D is a point in the base BC of an isosceles

A ABC. Prove that AB2=AD2+ BD. CD.

DP

(Let O be mid-point of BC, and suppose that D lies between B and O. Then BD=BO-OD, CD=CO+OD=BO+OD.)

Fig. 70.

Ex. 41. ABC is an isosceles triangle (AB=AC); BN is an altitude. Prove that 2AC.CN=BC2.

Ex. 42. BE, CF are altitudes of an acute-angled ▲ ABC. Prove that AE. AC=AF. AB. [Write down two different expressions for BC2.]

Ex. 48. In the figure of Ex. 42, prove that BC2=AB. FB+AC, EC.

Ex. 44. Prove that, if the sum of the squares on two opposite sides of a quadrilateral is equal to the sum of the squares on the two remaining sides, the diagonals of the quadrilateral must be at right angles.

Ex. 45. ABC is a triangle right-angled at A and from AB a part AD is marked off equal to AC. Show that the differences of the squares on BC and BD is equal to twice the rectangle AB. AC.

Ex. 46. ABC is an equilateral triangle and D is any point in the side BC. Prove that BC2=BD. DC + AD2.