EXAMINATIONS IN PLANE GEOMETRY FOR

ADMISSION TO HARVARD

JUNE, 1890

1. Prove that if two sides of a triangle are respectively equal to two sides of another, but the included angle in the first triangle is greater than the included angle in the second, the third side of the first triangle is greater than the third side of the second.

If the included angle in the case of the first triangle is twice as great as the included angle of the other, is the third side of the one twice as long as the third side of the other?

2. Prove that through three points not in the same straight line one circumference can be drawn and only

one.

A certain equilateral triangle has sides 83 inches long; what is the radius of the circumference circumscribed about this triangle?

3. Prove that an angle which has its vertex within a circumference is measured by half the sum of the two arcs intercepted between its sides when extended in both directions.

The vertices of a quadrilateral inscribed in a circle divide the circumference into arcs which are to each other as 1, 2, 3, and 4. Find the angles between the opposite sides of the quadrilateral.

4. Prove that the diagonals and the line which joins the middle points of the parallel sides of a trapezoid meet in a point.

5. The radius of a certain circle is 9 inches; find the area of that one of all the regular polygons inscribed in it which has the shortest perimeter. How long a perimeter can a regular polygon inscribed in this circle have?

JUNE, 1891

(In solving problems use for the approximate value 34.)

1. Prove that in an isosceles traingle the angles opposite the equal sides are equal to each other.

The area of a certain isosceles triangle is 50 square feet and each of its equal sides is 10 feet long; find the angles of the triangle.

2. Prove that in any quadrilateral circumscribed about a circle the sum of two opposite sides is equal to the sum of the other two opposite sides.

One vertex of a circumscribed quadrilateral and the directions of the sides which meet at the vertex are given; what is the locus of the centre of the inscribed circle?

3. From a fixed point 0 of a given circumference are drawn two chords OP, OQ, so as to make equal angles with a fixed chord, OR, between them. Prove that PQ will have the same direction whatever the magnitude of the equal angles.

4. The diagonals of a certain trapezoid which are 8 and 12 feet long, respectively, divide each other into segments which in the case of the shorter diagonal are 3 feet and 5 feet long. What are the segments of the other diagonal?

5. Assuming that as the number of sides of a circumscribed polygon is indefinitely increased, the perimeter approaches as a limit the circumference of the circle, and the area of the polygon the area of the circle; prove that

the area of a circle is numerically equal to one-half the product of its radius by its circumference.

6. A regular hexagon, the perimeter of which is 42 inches, is inscribed in a circle; what is the area of this circle?

JUNE, 1892

(In solving problems use for the approximate value 34.)

1. Prove that if two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less side.

In a certain right triangle one of the legs is half as long as the hypotenuse; what are the angles of the triangle?

2. Show how to find on a given indefinitely extended straight line in a plane, a point O which shall be equidistant from two given points A, B, in the plane. If A and B lie on a straight line which cuts the given line at an angle of 45° at a point 7 inches distant from A and 17 inches from B, show that OA will be 13 inches.

3. Prove that an angle formed by a tangent and a chord drawn through its point of contact is the supplement of any angle inscribed in the segment cut off by the chord. What is the locus of the centre of a circumference of given radius which cuts at right angles a given circumference?

4. Show that the areas of similar triangles are to each other as the squares of the homologous sides.

5. Prove that the square described upon the altitude of an equilateral triangle has an area three times as great as that of a square described upon half of one side of the triangle.

6. Find the area included between a circumference of radius 7 and the square inscribed within it.

JUNE, 1893

(In solving problems use for л the approximate value 34.)

1. Prove that two oblique lines drawn from a given point to a given line are equal if they meet the latter at equal distances from the foot of the perpendicular dropped from the point upon it.

How many l'nes can be drawn through a given point in a plane so as to form in each case an isosceles triangle with two given lines in the plane?

2. Prove that in the same circle, or in equal circles, equal chords are equally distant from the centre, and that of two unequal chords the less is at the greater distance from the centre.

Two chords of a certain circle bisect each other. One of them is 10 inches long; how far is it from the centre of the circle?

A variable chord passes, when produced, through a fixed point without a given circle. What is the locus of

the middle point of the chord?

3. A common tangent of two circumferences which touch each other externally at A, touches the two circumferences at B and C respectively; show that BA is perpendicular to AC.

4. Assuming that the areas of two triangles which have an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles, prove that the bisector of an angle of the triangle divides the opposite side into parts which are proportional to the sides adjacent to them.

5. Prove that the circumferences of two circles have the same ratio as their radii.

6. A quarter-mile running track consists of two parallel