LECTURE XV. (PROPOSITIONS 35-41). 1. Any point P is taken in the line joining an angular point A of a triangle to the middle point of the opposite side BC. Prove that the triangles APB, APC are equal. 2. ABC is a triangle, BD bisects AC in D, CE bisects AB in E, BD and CE intersect in F. Prove that the quadrilateral ADFE is equal to the triangle FBC. 3. If in the base of a triangle ABC there be taken any two points P and Q, equidistant from the extremities of the base, and if through each of these points two lines be drawn parallel to the sides AB, AC, so as to form two parallelograms having PA and QA for diagonals, show that these parallelograms are equal. 4. Two triangles of equal area are on the same base and opposite sides of it. Prove that the straight line joining the vertices is bisected by the base, or the base produced, and conversely if the straight line be bisected, the triangles are of equal area. 5. Bisect a triangle by a line drawn from a given point in one of the sides. 6. Bisect a parallelogram by a line drawn from a given point, (a) within, or (B) without it. 7. Every triangle is the half of a rectangle having the same base and altitude. 8. Show that the four triangles into which the diagonals divide a parallelogram are equivalent. 9. If one side of a triangle be bisected and a straight line be drawn from the point of bisection parallel to the base, the third side is bisected by it, and the triangle cut off by it is one-fourth of the given triangle. 10. Find the locus of the vertices of equivalent triangles upon the same base, and on the same side of it. LECTURE XVI. (PROPOSITIONS 42-45). 1. Describe a rhombus whose adjacent sides shall be to each other as 1 to 4. 2. State the process by which the area of any polygon may be converted into an equivalent rectangle. 3. Construct a rhombus equal to a given parallelogram. 4. Can a triangle be equal to a rectangle? If so, draw a rectangle equal to a given triangle. 5. Describe a triangle equal to a given rectangle. 6. Construct a triangle equal to any given quadrilateral figure. 7. One side of a rectangle being given, find the adjacent side, so that the rectangle may be equal to a given square. LECTURE XVII. (PROPOSITIONS 46–48). 1. The square on the side subtending an acute angle of a triangle is less than the squares on the sides containing the acute angle. 2. The square on the side subtending an obtuse angle of a triangle is greater than the squares on the sides containing the obtuse angle. 3. Demonstrate the converse of each of the preceding theorems. 4. The triangles ABC, DEF have the angles ABC, DEF right angles, and the sides AB, AC equal to DE, DF, each to each. Prove that the triangles are equal in every respect. 5. Being given the diagonal, construct the square. 6. Construct a square equal to the sum of two given squares. 7. Construct a square equal to the difference of two given squares. 8. Construct a square equal to three, four, or any number of squares. 9. Show how the squares on the sides of an isosceles rightangled triangle can be made to cover the square on the hypotenuse. 10. The parallelograms described on any two sides of a triangle are together equal to the parallelogram on its base having its side equal to and parallel to the line drawn from the vertex to the intersection of the produced exterior sides of the former figures. PROBLEMS ON THE CONSTRUCTION OF TRIANGLES. 1. Construct an isosceles triangle having the angles at the base together equal to half the vertical angle. 2. Construct an isosceles triangle having the angles at the base equal to thrice the vertical angle. 3. Construct a right-angled triangle having given the hypotenuse and the difference of the sides. 4. Construct a right-angled triangle having given the hypotenuse and the sum of the sides. 5. Construct a right-angled triangle having given the perimeter and an angle. 6. Construct a right-angled triangle having given the hypotenuse and the perpendicular from the right angle upon it. 7. Construct a triangle having given the perimeter and each of the angles. 8. Construct a triangle equivalent to a given triangle having its base in the same straight line as that of the given triangle, and its vertex in a given line parallel to the base. BOOK II. LECTURES XVIII., XIX., XX. (PROPOSITIONS 1-10). 1. Show, by diagram, that A (B+ C) = AB+ AC. 2. ABC is a triangle right-angled at C, D and E are points in AB and AB produced such that BD = BE = BC. Show that the rectangle contained by AD and AE is equal to the square on AC. 3. Each of the sides of a rectangle is double of the corresponding side of another rectangle. How many times does the larger rectangle contain the other? 4. Any rectangle is the half of the rectangle contained by the diameters of the squares on two adjacent sides. 5. If a straight line be divided into two equal and two unequal parts, the squares of the two unequal parts are equal to twice the rectangle contained by those parts, together with four times the square of the line between the points of section. 6. The square of the difference of two straight lines is less than the sum of the squares on the two lines by twice the rectangle contained by those lines. 7. The difference between the squares on any two straight lines is equal to the rectangle contained by the sum and difference of those lines. 8. AB is divided into any two parts in C, and AC and BC are bisected in D and E. Show that the square on AE and three times the square on BE are equal to the square on BD, and three times the square on AD. 9. The square on the sum of two straight lines, together with the square on their difference, is equal to twice the squares on the two straight lines. 10. The square of any straight line drawn from the vertex of an isosceles triangle to the base is less than the square of a side of the triangle by the rectangle contained by the segments of the base. LECTURE XXI. (PROPOSITIONS 11–14). 1. If one angle of a triangle is of a right angle, the square of the side subtending it is equal to the sum of the squares of the sides containing it, together with the rectangle contained by those sides. 2. Produce a given line so that the rectangle contained by the whole line thus produced and the part produced shall be equal to a given square. 3. Straight lines which are equal and parallel have equal projections on any other straight lines. 4. Parallel straight lines which have equal projections on another straight line are equal; and equal straight lines which have equal projections on another straight line are equally inclined to that line. 5. Show that in every parallelogram the squares of the diagonals are together equal to the sum of the squares of all the sides. 6. Produce a line so that the sum of the squares on the whole line thus produced and on the part produced may be equal to three times the square on the given line. 7. If ABC is a triangle having the angles at B and C double of the angle at A, show that the square on AB = the square on BC, together with the rectangle contained by AC, CB. 8. Describe a rectangle equal to a given square, and having the sum of two of its adjacent sides equal to a given straight line. 9. Find a square equal to a given equilateral triangle. 10. The sum of the squares on any two sides of a triangle is equal to twice the sum of the squares on half the base, and on the line joining the vertical angle with the point of bisection of the base. QUESTIONS SELECTED FROM THE EXAMINA- 1. The longest side of a given quadrilateral is opposite the shortest; prove that either of the angles adjacent to the shortest side is greater than the opposite angle of the quadrilateral. 2. Show how to determine a line, cut as in the proposition (II. 11), when the greater segment is given. 3. Find a point within a triangle from which lines drawn to the angular points will divide the triangle into three equal parts. 4. The angle C of a triangle ABC is a right angle, CD is the perpendicular on AB, and E is the middle point of AB; prove that the squares on AE, ED, DC, are together equal to half the square on AB. 5. Describe an isosceles triangle having each of the angles at the base double of the third angle. 6. Can triangles be equal which cannot be made to coincide Can squares or rectangles? Illustrate your answer by figures. 7. Show how to make a square equal to half a given square, 8. If a straight line be divided into two parts, the difference of the squares of the two parts is equal to the difference between the square on the whole line, and twice the rectangle contained by the whole and the lesser part. 9. Describe a square that shall be equal to the sum of two or more rectilineal figures. 10. Distinguish between a postulate and an axiom, and explain the necessity of a set of postulates and axioms as the basis of geometrical reasoning. 11. ABC is an isosceles triangle and in BA and CA produced points F and G are taken such that AF and AG are equal; prove that the angles AFC and AGB are equal. |