THEOREM 11. The ratio of the areas of similar polygons is equal to the ratio of the squares on corresponding sides. Construction In ABCDE take any point O. Let X be the corresponding point in PQRST. IV. 10. Ex. 1788. What is the ratio of the area of a room to the area by which it is represented on a plan whose scale is 1 in. to 1 ft.? Ex. 1789. On a map whose scale is 1 mile to 1 in., a piece of land is represented by an area of 20 sq. in.; what is the area of the land? Ex. 1790. On a map whose scale is 2 miles to 1 in., a piece of land is represented by an area of 24 sq. in.; what is the area of the land? Ex. 1791. What is the acreage of a field which is represented by an area of 3 sq. in. on a map whose scale is 25 in. to the mile? (640 acres =1 sq. mile.) Ex. 1792. What areas represent a field of 1 acre on maps in which 1 mile is represented by (i) 1 in., (ii) 1⁄2 in., (iii) 6 in., (iv) 2·5 in.? If the field were square, what would be the length of a line representing a side of the field? Ex. 1793. Two similar windows are glazed with small lozenge-shaped panes of glass, these panes being all identical in size and shape. The heights of the windows are 10 ft. and 15 ft. The number of panes in the smaller window is 1200; what is the number in the larger? Ex. 1794. A figure described on the hypotenuse of a right-angled triangle is equal to the sum of the similar figures described on the sides of the triangle. (This is a generalisation of Pythagoras' theorem.) Ex. 1795. Similar figures are described on the side and diagonal of a square; prove that the ratio of their areas is 1: 2. Ex. 1796. Similar figures are described on the side and altitude of an equilateral triangle; prove that the ratio of their areas is 4: 3. To construct a figure equivalent to a given figure A and similar to another figure B. Construction Reduce both figures to squares (see p. 333). Let / be a side of the figure B. Construct a length x so that b: a = l : x. On x describe a figure C similar to B; the side x of C corresponding to the side of B. Proof The area of C area of B = x2 : 12 = a2 : b2 = area of A area of B, .. area of C = area of A. Ex. 1797. square. Ex. 1798. triangle. Ex. 1799. Construct an equilateral triangle equivalent to a given Construct an equilateral triangle equivalent to a given Construct a rectangle having its sides in a given ratio and equivalent to a given square. Ex. 1800. MISCELLANEOUS EXERCISES. One of the parallel sides of a trapezium is double the other; show that the diagonals trisect one another. Ex. 1801. A straight line drawn parallel to the parallel sides of a trapezium divides the other two sides (or those sides produced) proportionally. Ex. 1802. Find the locus of a point which moves so that the ratio of its distances from two intersecting straight lines is constant. Ex. 1803. Through a given point within a given angle draw a straight line to be terminated by the arms of the angle, and divided in a given ratio (say) at the given point. Ex. 1804. Prove that two medians of a triangle trisect one another. Hence prove that the three medians pass through one point. Ex. 1805. The bisectors of the equal angles of two similar triangles are to one another as the bases of the triangles. G. S. II. 23 Ex. 1806. In two similar triangles, the parts lying within the triangle of the perpendicular bisectors of corresponding sides have the same ratio as the corresponding sides of the triangle. Ex. 1807. ABC, DEF are two similar triangles; P, Q are any two points in AB, AC; X, Y are the corresponding points in DE, DF. Prove that PQ:XY=AB: DE. Ex. 1808. The sides AC, BD of two triangles ABC, DBC on the same base BC and between the same parallels meet at E; prove that a parallel to BC through E, meeting AB, CD, is bisected at E. Ex. 1809. Show how to divide a parallelogram into five equivalent parts by lines drawn through an angular point. Ex. 1810. Divide a given line into two parts such that their mean proportional is equal to a given line. Is this always possible? Ex. 1811. Construct a rectangle equivalent to a given square, and having its perimeter equal to a given line. [See Ex. 1810.] Ex. 1812. A common tangent to two circles cuts the line of centres externally or internally in the ratio of the radii. Ex. 1813. On a given base construct a triangle having given the vertical angle and the ratio of the two sides. Ex. 1814. Construct a triangle having given the vertical angle, the ratio of the sides containing the angle, and the altitude drawn to the base. Ex. 1815. TP, TQ are tangents to a circle whose centre is C, CT cuts PQ in N; prove that CN. CT=CP2. Ex. 1816. In fig. 318, prove that ▲ PBC: A PAD BC: AD2. Is the same property true for fig. 319? Ex. 1817. In fig. 318, prove that PB. PC: PA. PD = BC2: AD2. Ex. 1818. ABCDE is a regular pentagon; BE, AD intersect at F; prove that EF is a third proportional to AD, AE. Ex. 1819. In fig. 295, the area of the regular hexagon obtained by joining the vertices of the star is three times that of the small hexagon. Ex. 1820. In fig. 319, PQ is drawn parallel to AD to meet BC produced in Q; prove that PQ is a mean proportional between QB, QC. Ex. 1821. The angle BAC of a AABC is bisected by AD, which cuts BC in D; DE, DF are drawn parallel to AB, AC and cut AC, AB at E, F respectively. Prove that BF: CE=AB2: AC2. Ex. 1822. ABC is a triangle right-angled at A; AD is drawn perpendicular to BC and produced to E so that DE is a third proportional to AD, DB; prove that ▲ ABD = ▲ CDE, and ▲ ABD is a mean proportional between ▲ ADC, BDE. Q, R are the points of Ex. 1823. Two circles touch externally at P; contact of one of their common tangents. Prove that QR2 is a mean proportional between their diameters. [Draw the common tangent at P, let it cut QR at S; join S to the centres of the two circles.] Ex. 1824. Two church spires stand on a level plain; a man walks on the plain so that he always sees the tops of the spires at equal angles of elevation. Prove that his locus is a circle. Ex. 1825. The rectangle contained by two sides of a triangle is equal to the square on the bisector of the angle between those sides together with the rectangle contained by the segments of the base. [See Ex. 1717.] Ex. 1826. The tangent to a circle at cuts two parallel tangents at Q, R; prove that the rectangle QP. PR is equal to the square on a radius of the circle. Ex. 1827. ABCD is a quadrilateral. If the bisectors of ▲ A, C meet on BD, then the bisectors of ▲ B, D meet on AC. Ex. 1828. Prove the validity of the following method of solving a quadratic equation graphically: Suppose that ax2 + bx + c =0 is the equation; on squared paper, mark the origin, from OX cut off OP = a, from P draw a perpendicular PQ upwards of length b, from Q draw to the left QR: = c(regard must be paid to the signs of a, b, c; e.g. if b is negative PQ will be drawn downwards); on OR describe a semicircle cutting PQ at S, T; the roots of the equation are |