A Second primer: plane geometry for preRMO and RMO: basic questions set

Problem 1:

If from any point in the bisector of an angle a straight line is drawn parallel to either arm of the angle, the triangle thus formed is isosceles.

Problem 2:

From X, a point in the base BC of an isosceles triangle ABC. a straight line is drawn at right angles to the base, cutting AB in Y, and CA produced in Z, show the triangle AYZ is isosceles.

Problem 3:

If the straight line which bisects an exterior angle of a triangle is parallel to the opposite side, show that the triangle is isosceles.

Problem 4:

The straight line drawn from any point in the bisector of an angle parallel to the arms of the angle, and terminated by them, are equal, and the resulting figure is a quadrilateral having all its sides equal.

Problem 5:

AB and CD are two straight lines intersecting at D, and the adjacent angles so formed are bisected; if through any point X in DC a straight line XYZ is drawn parallel to AB and meeting the bisectors Y and Z, show that XY is equal to XZ.

Problem 6:

Two straight rods PA, QB revolve about pivots at P and Q, PA making 12 complete revolutions a minute, and QB making 10. If they start parallel and pointing the same way, how long will it be before they are again parallel (i) pointing opposite ways (ii) pointing the same way?

Problem 7:

Prove that: if two straight lines are perpendicular to two other straight lines, each to each, the acute angle between the first pair is equal to the acute angle between the second pair.

Problem 8:

Show that the only regular figures which may be fitted together so as to form a plane surface are (i) equilateral triangles (ii) squares (iii) regular hexagons.

Problem 9:

If one side of a regular hexagon is produced, show that the exterior angle is equal to the interior angle of an equilateral triangle.

Problem 10:

If a straight line meets two parallel straight lines, and the two interior angles on the same side are bisected, show that the bisectors meet at right angles.

Problem 11:

If the base of any triangle is produced both ways, show that the sum of the two exterior angles minus the vertical angle is equal to two right angles.

Problem 12:

In the triangle ABC, the base angles at B and C are bisected by BO and CO respectively. Show that the angle BOC is 90 degrees plus A/2.

Problem 13:

In the triangle ABC, the sides AB, AC are produced, and the exterior angles are bisected by BO and CO. Show that the angle BOC is 90 degrees minus A/2.

Problem 14:

Prove: the angle contained by the bisectors of two adjacent angles of a quadrilateral is equal to half the sum of the remaining angles.

Problem 15:

A is the vertex of an isosceles triangle ABC, and BA is produced to D, so that AD is equal to BA; if DC is drawn, show that BCD is a right angle. Prove this.

Problem 16:

Prove: The straight line joining the middle point of the hypotenuse at a right angled triangle to the right angle is equal to half the hypotenuse.

Problem 17:

If two triangles have two angles of one equal to two angles of the other, each to each, and any one side of the first equal to the corresponding side of the other, the triangles are equal in all respects. (ASA congruency test):

Problem 18:

Show that the perpendiculars drawn from the extremities of the base of an isosceles triangle to the opposite sides are equal.

Problem 19:

Prove: Any point on the bisector of an angle is equidistant from the arms of the angle.

Problem 20:

Through O the middle point of a straight line AB, any straight line is drawn, and perpendiculars AX and BY are dropped upon it from A and B: show that AX is equal to BY.

Problem 21:

If the bisector of the vertical angle of a triangle is at right angles to the base, the triangle is isosceles.

Problem 22:

If in a triangle the perpendicular from the vertex on the base bisects the base, then the triangle is isosceles. Prove.

Problem 23:

If the bisector of the vertical angle of a triangle also bisects the base, the triangle is isosceles. Prove this.

Problem 24:

The middle point of any straight line which meets two parallel straight lines, and is terminated by them, is equidistant from the parallels. Prove this.

Problem 25:

A straight line drawn between two parallels and terminated by them is bisected; show that any other straight line passing through the middle point and terminated by the parallels is also bisected at that point.

Problem 26:

If through a point equidistant from two parallel straight lines, two straight lines are drawn cutting the parallels, the portions of the latter thus intercepted are equal. Prove this.

Problem 27:

In a quadrilateral ABCD, if AB=CD, and BC=DC: show that the diagonal AC bisects each of the angles which it joins, and that AC is perpendicular to BD.

Problem 28:

A surveyor wishes to ascertain the breadth of a river which he cannot cross. Standing at a point A, near the bank, he notes an object B immediately opposite on the other bank. He lays down a line AC of any length at right angles to AB, fixing a mark at O the middle point of AC. From C he walks along a line perpendicular to AC until he reaches a point D from which O and B are seen in the same direction. He now measures CD: prove that the result gives him the width of the river.

More later,

Cheers,

Nalin Pithwa.

PS: There is no royal road to geometry —- Plato.

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