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8. To construct a parallelogram equivalent to a given square and having the sum of its base and altitude equal to a given line. (Solve and prove.)

VIII.

1. Prove that only one perpendicular can be drawn from a point to a straight line.

2. Prove that of two sides of a triangle that is the greater which is opposite the greater angle. State and prove the converse.

3. Through a given point to draw a tangent to a given circle.

4. Prove that if a line be drawn so as to divide two sides of a triangle into proportional parts, this line is parallel to the third side.

5. To inscribe in a circle a regular decagon.

6. Prove that a triangle is equivalent to half of any parallelogram of the same base and altitude.

7. To find a triangle equivalent to a given polygon.

8. To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.

IX.

1. Prove that when oblique lines are drawn from a point in a perpendicular to points unequally distant from the foot of the perpendicular, the more remote line is the longer.

2. To bisect a given angle.

3. Draw a number of lines radiating from a point, and then draw two parallel lines intersecting them: prove that the parts of these parallels are proportional.

4. A tangent and a secant being drawn from a point outside a circle, prove that the tangent is a mean proportional between the entire secant and its exterior part.

5. What is the centre of a regular polygon? Prove that the sides of a regular polygon are equally distant from the centre.

6. The circumference of a circle is 341.8 feet; what is the circumference of another circle having twice the area of the former? (If you have not time to perform the computation, you can explain how to do it.)

X.

1. In what three cases is it proved that two triangles are equal? In what three cases, that they are similar? Define similar polygons.

2. Prove that if two opposite sides of a quadrilateral are equal and parallel, the other two sides are also equal and parallel. Define a Trapezoid

3. Prove that if two polygons are composed of the same number of triangles which are respectively similar and similarly disposed, the polygons are similar.

4. State and prove the theorem concerning the ratio between the areas of two similar triangles.

5. Prove that two regular polygons of the same number of sides are similar.

6. Find the formula for the area of a circle in terms of the radius and the ratio of the circumference to the diameter.

XL

1. To how many right angles is the sum of all the interior angles of any polygon equal? State and prove; and then state and draw the figure for the theorem on which this one immediately depends.

2. What is the measure of the angle formed by two chords which cut each other between the centre and the circumference? by two chords which meet at the circumference? by two secants which meet without the circumference? Draw the figure for each case, and prove the last one.

3. To describe a circle through three given points.

4. Prove that two regular polygons of the same number of sides are similar.

5. The area of a trapezoid is half the product of its altitude by the sum of its parallel sides.

6. The perimeter of a regular hexagon is 18. Find (a.) The area of the circumscribed circle;

(b.) The area of the square inscribed in this circle.

7. Prove the proportion that exists between the parts of two intersecting chords.

XII.

1. Two parallel lines are cut by a third line. Prove what angles formed by these lines are equal, and also what angles are supplements of each other.

2. Obtain the value of any interior angle of a regular octagon.

3. An angle inscribed in a circle is measured by half the arc intercepted by its sides. Prove this proposition for each of the three cases which may arise.

4. State and prove the method of finding the centre of a given circle or arc.

5. State and prove the method of finding a mean proportional between two given straight lines.

6. From a point without a circle secants are drawn to the circle. Prove the proportion existing between the entire secants and the parts lying outside the circle.

What corollary results when one of these secant lines becomes a tangent.

7. Show how a square may be constructed equal in area to any given polygon.

XIII.

1. The perimeters of similar polygons are to each other in what ratio? The areas of similar polygons are to each other in what ratio? Proof in both cases.

2. To make a square which is to a given square in a given ratio.

3. Prove that two rectangles are to each other as "the products of their bases by their altitudes. What follows if we suppose one of the rectangles to be the unit of surface?

4. Prove that two similar polygons may be divided into the same number of triangles, that are similar each to each and similarly placed.

5. To divide this line

into three parts proportional to the numbers 2, 4, and 3, and prove the principle involved.

6. Prove that a line which divides two sides of a triangle proportionally is parallel to the third side.

7. Prove that a tangent to a circle is perpendicular to the radius drawn to the point of contact.

8. Prove that parallel chords intercept upon the circumference equal arcs.

XIV.

1. Prove that two triangles are equal when a side and the two adjacent angles of the one are respectively equal to a side and the two adjacent angles of the other. Under what other conditions are two triangles equal to each other?

2. Prove that the diagonals of a parallelogram mutually bisect each other. Prove at what angle the diagonals of a rhombus bisect each other.

3. Given the circumference of a circle, show how to find the centre. Show also how to draw a tangent to the circumference, either from a point on the circumference or from one without it. Give the proof in the last case.

4. Prove that the area of any circumscribed polygon is half the product of its perimeter by the radius of the inscribed circle.

5. Show how a regular hexagon may be inscribed in a circle; also an equilateral triangle. Find the ratio of the side of the inscribed equilateral triangle to the radius of the circle.

6. Prove that similar triangles are to each other as the squares of their homologous sides.

7. Show how to find a triangle equivalent to a given polygon.

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