3. If in any triangle a median be perpendicular to the side to which it is drawn, the triangle shall be isosceles. Use Euc. I. 4 to show that A ADB A ADC. B DEFINITION.-The straight line drawn from a vertex of a triangle perpendicular to the opposite side, and terminated by that side, is called the Altitude from that vertex. Thus in Ex. 3, AD is both median and altitude. 4. AK, the bisector of the A of the triangle ABC, meets BC in K; show that AB>BK and AC> CK. Hence obtain a second proof of Euc. I. 20. Use Euc. I. 16 to show that LAKB > < KAC (or & KAB), and apply Euc. I. 19. B K DEFINITION.--In a right-angled triangle the side opposite to the right angle is called the Hypotenuse. 5. In any right-angled triangle, the hypotenuse is greater than either of the sides containing the right angle. Use Euc. I. 17 and I. 19. B A 6. In any obtuse-angled triangle the side opposite to the obtuse angle is greater than either of the sides containing the obtuse angle. B A Use Euc. I. 17 and I. 19. 7. ABC is a triangle in which AB> AC and P is any point in BC. Show that AB > AP. Р Use Euc. I. 18 to show that LACB LABC, and Euc. I. 16 to show that LAPB><ACB, etc., and apply Euc. I. 19. DEFINITION. The sum of the sides of any rectilineal figure is called its Perimeter. B Thus BC+CA+AB is the perimeter of the A ABC. 8. The sum of the medians of a triangle is less than the perimeter of the triangle. A C Use Ex. 1 for the three medians in turn, add the results, and divide by 2. (See note on § 1, Ex. 4, as to the use of standard theorems.) 9. Of the three exterior angles of a triangle, any two are together greater than two right angles. Use Euc. I. 17. 10. In the quadrilateral ABCD, AB is the longest and CD the shortest side. Show that <C> < A and < D > < B. § 6. (Bookwork, EUCLID, I. 1-21.) 1. If any point be taken within a triangle, the sum of its distances from the angular points shall be (1) greater than half the perimeter of the triangle, and (2) less than the perimeter. Let P be any point within ▲ ABC; it is required to prove that (1) PA+PB+PC>(BC+CA+AB), (2) PA+PB+PC<BC+CA+AB. 2. The perimeter of a quadrilateral is (1) greater than the sum of the diagonals, and (2) less than twice the sum of the diagonals. AB+BC>AC, BC+CD>BD, etc.; ... 2(AB+BC+CD+DA)>2(AC+BD), etc. For (2) use Euc. I. 20 to show that OA+OB>AB, etc. 3. Any three sides of a quadrilateral are together greater than the fourth side. Draw a diagonal and use Euc. I. 20. 4. If P be a point within a quadrilateral ABCD, Produce AP to meet BC or CD, and use Euc. I. 20 and the method of Ex. 3 to show in fig. 1 that BC+CD+DA>AE+EC+CB>AP+PB, and in fig. 2 that BC+CD+DA>AF+FB, etc. 5. If a point be taken within a quadrilateral, the sum of its distances from the angular points shall be (1) greater than one-half, and (2) less than three-halves of the perimeter. A For (1) use Euc. I. 20. PA+PB>AB, etc. For (2) show, as in Ex. 4, that BC+CD+DA>PA+PB, etc. 6. If within a quadrilateral ABCD, P be a point which is not the point of intersection of the diagonals, PA+PB+PC+PD>AC+BD. Use Euc. I. 20. C 7. The difference between two sides of a triangle is less than the third side. B From AB cut off AD=AC. Join CD. Use Euc. I. 16 to show that < BDC>LACD and ADC> 4 BCD. [Euc. I. 20. Apply Euc. I. 5 and I. 19 to deduce BC>BD. Otherwise. AC+BC>AB, or AC+BC>AD+BD. But AC AD; .:. BC>BD. B |