THEOREM 28. If a triangle and a parallelogram stand on the same base and between the same parallels, the area of the triangle is half that of the parallelogram.

To construct a triangle equivalent to a given quadrilateral.

Let ABCD be the given quadrilateral.

Construction. Join CA. Through D draw

DD' || to CA, meeting BA produced in D'.

Ex. 5. Prove that the area of a rhombus is half the product of its diagonals.

Ex. 6. F, E are the mid-points of the sides AD, BC of a parallelogram ABCD; P is any point in FE. Prove that ▲ APB = ▲ ECF.

Ex. 7. ABCD is a parallelogram; P, Q the mid-points of AB, AD. Prove that ▲ APQ= of ABCD. (Join PD, BD.)

Ex. 8. AD is a median of the triangle ABC. Prove ▲ ABD = ▲ ACD.

Ex. 9. The base BC of A ABC is divided at D so that BDBC; prove that ▲ ABD=ACD.

Ex. 10. The ratio of the areas of triangles of the same height is equal to the ratio of their bases.

Ex. 11. The ratio of the areas of triangles on the same base is equal to the ratio of their heights.

Ex. 12. ABCD is a quadrilateral and the diagonal AC bisects the diagonal BD. Prove that AC divides the quadrilateral into equivalent triangles (fig. 51).

Ex. 13. If ABCD is divided into equal areas by the diagonal AC, prove that BD is bisected by AC.

Ex. 14. E is a point on the median AD of ▲ ABC; prove that ▲ ABE A ACE.

B

OR

Fig. 51.

Ex. 15. A four-sided field is to be divided into two parts of equal area; prove the accuracy of the following construction. Draw a quadrilateral ABCD to represent the field; draw the diagonal AC; find E, the mid-point of AC; join BE, DE; then the areas ABED and CBED are equal. Ex. 16. D is a point on the base BC of AABC; E is the mid-point of AD; prove that ▲ EBC = ABC.

¶Ex. 17. Draw a triangle PQR; mark X, Y the mid-points of PQ, PR; join XY, QY, RX. By Th. 23, p. 46, we know that XY is || to QR. (i) Name two equivalent triangles on the base QR. (ii) Name two equivalent triangles on the base XY. (iii) Name two equivalent triangles with X as their common vertex. (iv) Name two equivalent triangles with Y as their common vertex. (v) Prove that As PXR, PYQ are equivalent. (vi) If QY, RX intersect at O, prove that As OQX, ORY are equivalent.

Ex. 18. PA is parallel to BC.

PC and AB meet in O. Prove that
Δ ΡΟΒ - Δ AOC.

=

Ex. 19. A line parallel to the base BC of AABC cuts the sides AB, AC in D, E respectively. Prove that AABE A ACD.

=

Ex. 20. F is any point on the base BC of ▲ ABC; E is the mid-point of BC. ED is drawn parallel to AF. Prove that

=

ADFC AABC. (See fig. 52.)

Ex. 21. Draw a line through a given point of a side of a triangle to bisect the area of the triangle. (See Ex. 20.)

Ex. 22. O is a point inside a parallelogram ABCD; prove that

▲ OAB+ A OCD=ABCD.

Ex. 23. In fig. 53 ▲ PXQ=▲ RXS; prove that PR is parallel to QS.

Ex. 24. In fig. 54 ▲ AEB= ▲ ADC; prove that DE is parallel to BC.

Ex, 25. ABCD is a quadrilateral, and ABCF a parallelogram; prove that, if the diagonal AC bisects the quadrilateral, DF is parallel to AC.

Ex. 26. Show how to divide a triangle into four equivalent triangles.

¶Ex. 27. Draw a scalene triangle, and transform it into an equivalent isosceles triangle on the same base. (Keep the base fixed; where must the vertex be in order that the triangle may be isosceles? Where must the vertex be in order that the triangle may be equivalent to the given triangle?)

TEx. 28. Show how to transform a given triangle

(i) into an equivalent right-angled triangle.

(ii) into an equivalent triangle on the same base, having one side of 2 inches. Is this always possible?

(iii) into an equivalent triangle having one angle equal to a given angle.

(iv) into an equivalent right-angled triangle with one of the sides about the right angle equal to 5 cm. (First make one side 5 cm.; then take this as base and make the triangle right-angled.)

(v) into an equivalent isosceles triangle with base equal to a given line. Ex. 29. Transform an equilateral triangle of sides 3 in. into an equivalent triangle with a side of 4 in., and an angle of 60° adjacent to that side.

Ex. 30. Show how to transform a given triangle into an equivalent triangle with its vertex (i) on a given line, (ii) one inch from a given line, (iii) one inch from a given point, (iv) equidistant from two given intersecting lines.

Ex. 31. Show how to construct, on a base of given length, a triangle equivalent to a given triangle.

Ex. 32. Show how to construct a triangle having an altitude of given length and area equal to that of a given triangle.

Ex. 33. Show how to transform a given rectangle into an equivalent rectangle having one side equal to a given length.

Ex. 84. Given a quadrilateral ABCD, construct an equivalent triangle on base AB having LA in common with the quadrilateral.

Ex. 35. ABCD is a quadrilateral whose diagonals intersect at E. AC is produced to F so that CF AE. Prove that A FBD=the given quadrilateral.

Ex. 36. Prove that the area of a quadrilateral ABCD is equal to that of a triangle which has two sides equal to the diagonals and the contained angle equal to that between the diagonals. (See Ex. 35.)

Ex. 37. AB, CD are parallel sides of a trapezium ABCD; E is the mid-point of AD; prove that ▲ BEC= trapezium. (Through E draw a line parallel to BC cutting the two parallel sides.)

Ex. 38. P, Q are the mid-points of the sides BC, AD of the trapezium ABCD; EPF, GQH are drawn perpendicular to the base. Prove that trapezium=rectangle GF. (See fig. 55.)

Ex. 39. O is any point on the diagonal BD of a parallelogram ABCD. EOF, GOH are parallel to AB, BC respectively. Prove that parallelogram AO= parallelogram CO. (See fig. 56.)

Ex. 40. Any straight line drawn through the centre of a parallelogram (i.e. through the intersection of the diagonals) bisects the parallelogram.

Ex. 41. Show how to bisect a parallelogram by a straight line drawn perpendicular to a side.

Ex. 42. Show how to bisect a parallelogram by a straight line drawn through a given point.

Ex. 43. E is any point on the diagonal AC of a parallelogram ABCD. Prove that AABE = AADE.

Ex. 44. Produce the median BD of a triangle ABC to E, making DE=DB. Prove that ▲ EBC= ▲ ABC.

Ex. 45. L, M are the mid-points of the parallel sides AB, DC of a trapezium ABCD. Prove that LM bisects the trapezium.

Ex. 46. In Ex. 45, O is the mid-point of LM; prove that any line through O which cuts AB, CD (not produced) bisects the trapezium.

Ex. 47. Prove that the area of the parallelogram formed by joining the midpoints of the sides of any quadrilateral (see Ex. 44, p. 51) is half the area of the quadrilateral.

Ex. 48. The medians BD, CE of AABC intersect at G; prove that quadrilateral ADGE=▲ BGC. (Add to each a certain triangle.)

Ex. 49. Draw a quadrilateral ABCD; from B and C draw BE and CF perpendicular to AD. Prove that the area of the quadrilateral ABCD is equal to the sum of the areas of the triangles ABF and ECD.

[First consider the case in which angles A and D are both acute.]

Ex. 50. In fig. 57 ABCD is a square and DEGH is a rectangle. Prove that they are equal in area.

Ex. 51. ABCD is a quadrilateral with the side DC parallel to the side AB; O is the middle point of BC, and DO and AB when produced meet at E. Prove that the area of the quadrilateral ABCD is twice the area of the triangle EOA.

H

A

B

E

F

Fig. 57.

Ex. 52. P is any point on the side BC of a quadrilateral ABCD; BX is drawn parallel to AP, and CY is drawn parallel to DP. Prove that, if BX meets CY in Q, the area of the triangle QAD is equal to the area of the quadrilateral ABCD.

CHAPTER VIII

THE THEOREM OF PYTHAGORAS,
ILLUSTRATIONS OF IDENTITIES,
EXTENSIONS OF PYTHAGORAS

THE THEOREM OF PYTHAGORAS.

THEOREM 29. In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the sides containing the right angle.

Data

Fig. 58.

ABC is a triangle, right-angled at A.

The figures BE, CH, AF are squares described upon BC, CA AB respectively.

To prove that

sq. BE =sq. CH + sq. AF.

Construction Through A draw AL || to BD (or CE).