5. Show that the bisector of the exterior angle at the vertex of an isosceles triangle is parallel to the base.
6. A ladder erected against an inner wall of a shed just reaches a window 18 feet from the ground, and, on being turned over through a right angle (the foot not being moved), it reaches a point on the opposite wall 7 feet 6 inches from the ground. Find the distance between the walls.
7. D is the middle point of the hypotenuse BC of a right-angled triangle BAC. Show that DA = DB.
8. AB, CD are two equal chords in a circle. Show that they are equidistant from the centre.
9. Show that, if two tangents are drawn to a circle from an external point, the tangents (1) are equal in length, (2) subtend equal angles at the centre of the circle.
III. LATIN PROSE COMPOSITION.
Translate into Latin:—
When the bridge was nearly all cut away, Horatius made his two companions leave him, and pass over into the city. Then he stood alone on the bridge, and defied all the army of the Etruscans: and they showered their javelins upon him, and he caught them on his shield and stood yet unhurt. But just as they were rushing at him, to drive him from his post by main force, the last beams of the bridge gave way, and it all fell with a mighty crash into the river. While the Etruscans wondered, and stopped in their course, Horatius turned and prayed to the god of the river: ‘O father Tiber, I pray thee to receive these arms and me who bear them, and to let thy waters befriend and save me.’ Then he leaped into the river, and though the darts fell all around him, yet they did not wound him, and he swam across to the city safe and sound.