MISCELLANEOUS EXERCISES ON PLANE GEOMETRY

Ex. 1052. If equilateral triangles are constructed on the sides of any given triangle, the lines joining the vertices of the given triangle to the outer vertices of the opposite equilateral triangles are equal.

Ex. 1053. If, on the arms of a right triangle as diameters, semicircles are drawn so as to lie outside of the triangle, and if, on the hypotenuse as a diameter, a semicircle is drawn passing through the vertex of the right angle, the sum of the areas of the two crescents included between the semicircles is equal to the area of the given triangle.

Ex. 1054. The area of the regular inscribed triangle is half that of the regular inscribed hexagon.

Ex. 1055.

From a given point draw a secant to a circle such that its internal and external segments shall be equal.

Ex. 1056. Show that the diagonals of any quadrilateral inscribed in a circle divide the quadrilateral into four triangles which are similar, two and two.

Ex. 1057. Through a point P, outside of a circle, construct a secant PAB so that AB2 = PA × PB.

Ex. 1058. The radius of a circle is 6 feet. circles concentric with it whose circumferences equivalent parts?

Ex. 1059. Given parallelogram ABCD, F the mid-point of BC; prove OT =

What are the radii of the divide its area into three

TC.

Ex. 1060. Given PT a tangent to a circle at point T, and two other tangents parallel

A

B

F

C

0

T

D

to each other cutting PT at A and B respectively; prove that the radius of the circle is a mean proportional between AT and TB.

Ex. 1061. Show that a mean proportional between two unequal lines is less than half their sum.

Ex. 1062. Given two similar triangles, construct a triangle equivalent to their sum.

Ex. 1063. The square of the side of an inscribed equilateral triangle is equal to the sum of the squares of the sides of the inscribed square and of the inscribed regular hexagon.

Ex. 1064. Prove that the area of a circular ring is equal to the area of a circle whose diameter equals a chord of the outer circumference which is tangent to the inner.

Ex. 1065. If two chords drawn from a common point P on the circumference of a circle are cut by a line parallel to the tangent through P, the chords and the segments of the chords between the two parallel lines are inversely proportional.

Ex. 1066. Construct a segment of a circle similar to two given similar segments and equivalent to their sum.

Ex. 1067. The distance between two parallels is a, and the distance between two points A and B in one parallel is 2 b. Find the radius of the circle which passes through A and B, and is tangent to the other parallel.

Ex. 1068. Tangents are drawn through a point 6 inches from the circumference of a circle whose radius is 9 inches. Find the length of the tangents and also the length of the chord joining the points of contact. Ex. 1069. If the perimeter of each of the figures, equilateral triangle, square, and circle, is 396 feet, what is the area of each figure?

Ex. 1070. The lengths of two sides of a triangle are 13 and 15 inches, and the altitude on the third side is 12 inches. Find the third side, and also the area of the triangle. (Give one solution only.)

Ex. 1071. If the diameter of a circle is 3 inches, what is the length of an arc of 80° ?

Ex. 1072. AD and BC are the parallel sides of a trapezoid ABCD, whose diagonals intersect at E. If F is the mid-point of BC, prove that FE prolonged bisects AD.

Ex. 1073. Given a square ABCD. Let E be the mid-point of CD, and draw BE. A line is drawn parallel to BE and cutting the square. Let P be the mid-point of the segment of this line within the square. Find the locus of P when the line moves, always remaining parallel to BE. Describe the locus exactly, and prove the correctness of your answer.

Ex. 1074. Let ABCD be any parallelogram, and from any point P in the diagonal AC draw the straight line PM cutting AB in M, BC in N, CD in L, and AD in K. Prove that PM. PN PK · PL.

Ex. 1075. Find the area of a segment of a circle whose height is 4 inches and chord 8√3 inches.

Ex. 1076. A square, whose side is 5 inches long, has its corners cut off in such a way as to make it into a regular octagon. Find the area and the perimeter of the octagon.

Ex. 1077. Into what numbers of arcs less than 15 can the circumference of a circle be divided with ruler and compasses only?

Ex. 1078. Through a point A on the circumference of a circle chords are drawn. On each one of these chords a point is taken one third of the distance from A to the other end of the chord. Find the locus of these points, and prove that your answer is correct.

Ex. 1079. In what class of triangles do the altitudes meet within the triangle? on the boundary? outside the triangle? Prove.

Ex. 1080. Given a triangle ABC and a fixed point D on side AC; draw the line through D which divides the triangle into two parts of equal area.

Ex. 1081. The sides of a triangle are 5, 12, 13. Find the radius of the circle whose area is equal to that of the triangle.

Ex. 1082. In a triangle ABC the angle C is a right angle, and the lengths of AC and BC are 5 and 12 respectively; the hypotenuse BA is prolonged through A to a point D so that the length of AD is 4; CA is prolonged through A to E so that the triangles AED and ABC have equal areas. What is the length of AE?

Ex. 1083. Given three points A, B, and C, not in the same straight line; through A draw a straight line such that the distances of B and C from the line shall be equal.

Ex. 1084. Given two straight lines that cut each other; draw four circles of given radius that shall be tangent to both of these lines..

Ex. 1085. Construct two straight lines whose lengths are in the ratio of the areas of two given polygons.

Ex. 1086. The radius of a regular inscribed polygon is a mean proportional between its apothem and the radius of the similar circumscribed polygon.

Ex. 1087. Draw a circumference which shall pass through two given points and bisect a given circumference.

Ex. 1088. A parallelogram is constructed having its sides equal and parallel to the diagonals of a given parallelogram. Show that its diagonals are parallel to the sides of the given parallelogram.

HINT. Look for similar triangles.

Ex. 1089. If two chords are divided in the same ratio at their point of intersection, the chords are equal.

Ex. 1090. The sides AB and AC of a triangle ABC are bisected in D and E respectively. Prove that the area of the triangle DBC is twice that of the triangle DEB.

Ex. 1091. Two circles touch externally. How many common tangents have they? Give a construction for the common tangents.

Ex. 1092. Prove that the tangents at the extremities of a chord of a circle are equally inclined to the chord.

Ex. 1093. Two unequal circles touch externally at P; line AB touches the circles at A and B respectively. Prove angle APB a right angle.

Ex. 1094. Find a point within a triangle such that the lines joining this point to the vertices shall divide the triangle into three equivalent parts.

Ex. 1095. A triangle ABC is inscribed in a circle. The angle B is equal to 50° and the angle C is equal to 60°. What angle does a tangent at A make with BC prolonged to meet it?

Ex. 1096. The bases of a trapezoid are 8 and 12, and the altitude is 6. Find the altitudes of the two triangles formed by prolonging the nonparallel sides until they intersect.

Ex. 1097. The circumferences of two circles intersect in the points A and B. Through A a diameter of each circle is drawn, viz. AC and AD. Prove that the straight line joining C and D passes through B.

Ex. 1098. How many lines 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?

Ex. 1099. The lengths of two chords drawn from the same point in the circumference of a circle to the extremities of a diameter are 5 feet and 12 feet respectively. Find the area of the circle.

Ex. 1100. Through a point 21 inches from the center of a circle whose radius is 15 inches a secant is drawn. Find the product of the whole secant and its external segment.

Ex. 1101. The diagonals of a rhombus are 21 feet and 40 feet respectively. Compute its area.

Ex. 1102. On the sides AB, BC, CA of an equilateral triangle ABC measure off segments AD, BE, CF, respectively, each equal to one third the length of a side; draw triangle DEF; prove that the sides of triangle DEF are perpendicular respectively to the sides of triangle ABC.

=*; (b) x = a √5. х 3

Ex. 1104. Find the area included between a circumference of radius 7 and an inscribed square.

Ex. 1105. What is the locus of the center of a circle of given radius whose circumference cuts at right angles a given circumference?

Ex. 1106. Two chords of a certain circle bisect each other. One of them is 10 inches long; how far is it from the center of the circle?

Ex. 1107. Show how to find on a given straight line of indefinite length a point O which shall be equidistant from two given points A and 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.

Ex. 1108. A variable chord passes, when prolonged, through a fixed point outside of a given circle. What is the locus of the mid-point

of the chord ?

Ex. 1109. A certain parallelogram inscribed in a circle has two sides 20 feet in length and two sides 15 feet in length. What are the lengths of the diagonals?

Ex. 1110. Upon a given base is constructed a triangle one of the base angles of which is double the other. The bisector of the larger base angle meets the opposite side at the point P. Find the locus of P.

Ex. 1111. What is the locus of the point of contact of tangents drawn from a fixed point to the different members of a system of con1 centric circles?

Ex. 1112. Find the locus of all points, the perpendicular distances of which from two intersecting lines are to each other as 3 to 2.

Ex. 1113.

The sides of a triangle are a, b, c. Find the lengths of

the three medians.

Ex. 1114.

their sum.

Ex. 1115.

Given two triangles; construct a square equivalent to

In a circle whose radius is 10 feet, two parallel chords are drawn, each equal to the radius. Find the area of the portion between these chords.

Ex. 1116. A has a circular garden and B one that is square. The distance around each is the same, namely, 120 rods. Which has the more land, A or B ? How much more has he?

Ex. 1117. Prove that the sum of the angles of a pentagram (a fivepointed star) is equal to two right angles.

Ex. 1118. AB and A'B' are any two chords of the outer of two concentric circles; these chords intersect the circumference of the inner circle in points P, Q and P', Q' respectively prove that AP. PB=A'P' · P'B'.

Ex. 1119. A running track consists of two parallel straight portions joined together at the ends by semicircles. The extreme length of the plot inclosed by the track is 176 yards. If the inside line of the track is a quarter of a mile in length, find the cost of seeding this plot at cent a square yard. (2.)