Question 826466
There are two possibilities:<ul><li>Segment AD is perpendicular to to segment BC. If this is true, then triangle ADB is a right triangle (since perpendiculars form right angles). And AD is a leg of this triangle and AB is the hypotenuse. Hypotenuse's are always the longest side of a right triangle (since it is opposite the largest angle).</li><li>It is unknown if AD is perpendicular to BC. In this case:<ol><li>Draw a perpendicular from A to BC and label the point on BC as point E.</li><li>We now have two right triangles: AEB and AED.</li><li>Both right triangles share the leg AE.</li>The other legs of these triangles are EB for triangle AEB and ED for triangle AED.</li><li>EB > ED</li><li>AB is the hypotenuse of triangle AEB and AD is the hypotenuse of triangle AED.</li><li>Triangle AEB, with one leg identical and the other leg longer than triangle AED's, must have a longer hypotenuse.</li><li>So AB > AD.</li></ol></li></ul>Note: If point D is closer to point C than to point B then the logic above will not be as clear. But if you change the references from AB to AC then the logic still works. And, if AC > AD and AB = AC, then AB > AD.