EXERCISES ON THE AREA OF A TRIANGLE. {Theoretical.) 1. ABC is a triangle and XY is drawn parallel to the base BC, cutting the other sides at X and Y. Join BY`and CX; and shew that (i) the AXBC= the ▲ YBC; (ii) the ▲ BXY= the ACXY; (iii) the AABY = the ACX. If BY and CX cut at K, shew that (iv) the BKX=the ACKY. 2. Shew that a median of a triangle divides it into two parts of equal area. How would you divide a triangle into three equal parts by straight lines drawn from its vertex? Prove that a parallelogram is divided by its diagonals into four triangles of equal area. 4. ABC is a triangle whose base BC is bisected at X. point in the median AX, shew that the AABY= the AACY in area. If Y is any 5. ABCD is a parallelogram, and BP, DQ are the perpendiculars from B and D on the diagonal AC. Shew that BP=DQ. Hence if X is any point in AC, or AC produced, prove (i) the ▲ ADX = the ▲ ABX; (ii) the CDX= the CBX. 6. Prove by means of Theorems 26 and 27 that the straight line joining the middle points of two sides of a triangle is parallel to the third side. 7. The straight line which joins the middle points of the oblique Vsides of a trapezium is parallel to each of the parallel sides. 8. ABCD is a parallelogram, and X, Y are the middle points of the sides AD, BC; if Z is any point in XY, or XY produced, shew that the triangle AZB is one quarter of the parallelogram ABCD. 9. If ABCD is a parallelogram, and X, Y any points in DC and AD respectively shew that the triangles AXB, BYĊ are equal in area. 10. ABCD is a parallelogram, and P is any point within it; shew that the sum of the triangles PAB, PCD is equal to half the parallelogram. EXERCISES ON THE AREA OF A TRIANGLE. (Numerical and Graphical.) 1. The sides of a triangular field are 370 yds., 200 yds., and 190 yds. Draw a plan (scale 1" to 100 yards). Draw and measure an altitude; hence calculate the approximate area of the field in square yards. 2. Two sides of a triangular enclosure are 124 metres and 144 metres respectively, and the included angle is observed to be 45°. Draw a plan (scale 1 cm. to 20 metres). Make any necessary measure. ment, and calculate the approximate area. 3. In a triangle ABC, given that the area=6'6 sq. cm., and the base BC 5.5 cm., find the altitude. Hence determine the locus of the vertex A. If in addition to the above data, BA=2.6 cm., construct the triangle; and measure CA. 4. In a triangle ABC, given the altitude, and the locus of A. and measure b. 5. area=3'06 sq. in., and a=30". Find Given C-68°, construct the triangle; ABC is a triangle in which BC, BA have constant lengths 6 cm. and 5 cm. If BC is fixed, and BA revolves about B, trace the changes in the area of the triangle as the angle B increases from 0° to 180°. Answer this question by drawing a series of triangles, increasing B by increments of 30°. Find the area in each case and tabulate the results. (Theoretical.) 6. If two triangles have two sides of one respectively equal to two sides of the other, and the angles contained by those sides supplementary, shew that the triangles are equal in area. Can such triangles ever be identically equal? 7. Shew how to draw on the base of a given triangle an isosceles triangle of equal area. 8. If the middle points of the sides of a quadrilateral are joined in order, prove that the parallelogram so formed [see Ex. 7, p. 64], is half the quadrilateral. 9. ABC is a triangle, and R, Q the middle points of the sides AB, AC; shew that if BQ and CR intersect in X, the triangle BXC is equal to the quadrilateral AQXR. 10. Two triangles of equal area stand on the same base but on opposite sides of it: shew that the straight line joining their vertices is bisected by the base, or by the base produced. [The method given below may be omitted from a first course. any case it must be postponed till Theorem 29 has been read.] ln The Area of a Triangle. Given the three sides of a triangle, to calculate the area. EXAMPLE. Find the area of a triangle whose sides measure 21 m., 17 m., and 10 m. Find by the above method the area of the triangles, whose sides are as follows: 7. If the given sides are a, b and c units in length, prove (iii) ▲ = ‡√(a+b+c) ( − a+b+c) (a − b+c) (a + b − c). THE AREA OF QUADRILATERALS. THEOREM 28. To find the area of (i) a trapezium. (i) Let ABCD be a trapezium, having the sides AB, CD parallel. Join BD, and from C and D draw perpendiculars CF, DE to AB. Let the parallel sides AB, CD measure a and b units of length, and let the A height CF contain h units. Then the area of ABCD=▲ ABD+^ DBC 1 AB.DE+DC.CF h That is, 1 2 1 = ah+ bh="2 (a+b). the area of a trapezium height x (the sum of the parallel sides). = (ii) Let ABCD be any quadrilateral. Draw a diagonal AC; and from B and D draw perpendiculars BX, DY to AC. These perpendiculars are called offsets. If AC contains d units of length, and BX, DY p and q units respectively, A the area of the quad1 ABCD = ABC+^ADC 1 2 the area of a quadrilateral = diagonal × (sum of offsets). B L EXERCISES. (Numerical and Graphical.) 1. Find the area of the trapezium in which the two parallel sides are 4.7" and 3.3′′, and the height 1.5′′. 2. In a quadrilateral ABCD, the diagonal AC-17 feet; and the offsets from it to B and D are 11 feet and 9 feet. Find the area. 3. In a plan ABCD of a quadrilateral enclosure, the diagonal AC measures 8.2 cm., and the offsets from it to B and D are 3.4 cm. and 26 cm. respectively. If 1 cm. in the plan represents 5 metres, find the area of the enclosure. 4. Draw a quadrilateral ABCD from the adjoining rough plan, the dimensions being given in inches. Draw and measure the offsets to A and C from the diagonal BD; and hence calculate the area of the quadrilateral. 6. Draw a trapezium ABCD from the following data: AB and CD are the parallel sides. AB=4"; AD=BC=2"; the LA=the LB=60°. Make any necessary measurements, and calculate the area. 7. Draw a trapezium ABCD in which AB and CD are the parallel sides; and AB=9 cm., CD=3 cm., and AD=BC=5 cm. Make any necessary measurement, and calculate the area. 8. From the formula area of quad' diag. × (sum of offsets) shew that, if the diagonals are at right angles, area=} (product of diagonals). 9. Given the lengths of the diagonals of a quadrilateral, and the angle between them, prove that the area is the same wherever they intersect. |