81. Draw an obtuse-angled triangle. Circumscribe a circle about it.

82. Without using the compasses, draw an arc.

Can you find

the center of the circle of which the arc forms a part?

83. Construct a rhombus whose sides are 4 inches, the acute angle being 30°. What is its altitude?

84. Construct a square on a line 3 inches long, using only arcs of 3 inches radius.

85. Can you construct a triangle with sides 2, 4, and 6 inches? 86. Draw a circle, radius 3 inches. Draw an equilateral triangle around it.

87. Construct an equilateral triangle, altitude 14 inches. Construct one having an altitude twice as great.

88. Construct a triangle whose sides measure 1, 11, and 2 inches, respectively. Construct a second having its corresponding sides twice as large as those of the first triangle.

Construct a triangle whose angles measure 30°, 60°, and 90°, respectively. Try to construct a second triangle having its corresponding angles twice as large as those of the first triangle.

89. Construct a square which shall be equal in area to two squares, one having a side of 2 inches, the other having a side of 3 inches.

90. Can you construct a square whose area will be 13 square inches?

91. Construct a square whose area will be equal to the difference of area between two squares, one having a side of 3 inches, the other having a side of 2 inches.

92. Construct a square whose side is 3 inches. Construct another having double the area.

93. Construct an equilateral triangle, side 2 inches. How many 1-inch equilateral triangles can be made from it?

94. How many 1-inch equilateral triangles can be made from an equilateral triangle whose sides are 3 inches?

From one whose sides are 4 inches? 5 inches?

95. Construct a triangle, sides 2, 3, 4 inches. Divide it into four equal triangles. Give the dimensions of each of the latter.

96. Construct a triangle 1, 1, 2 inches. Make a triangle nine times as large as the first by producing two of the sides and drawing a fourth line. What are the dimensions of the second triangle?

97. Draw a circle of 1 inch radius. Tangent to it, and enclosing it, draw one having four times the area of the first.

98. Can you draw a circle having half the area of a circle whose radius is 2 inches?

99. Draw an equilateral triangle, side 2 inches. Construct another of half the area of the first.

100. Can you tell the ratio between the area of an inscribed and that of a circumscribed hexagon?

EQUAL TRIANGLES.

EQUIVALENT TRIANGLES.

1271. NOTE. - The protractor and the triangle may be used in the following exercises.

1. Draw a rectangle, base 2 inches, altitude 2 inches. Draw a rhomboid, base 24 inches, altitude 2 inches. Find the area of each.

2. With a base 2 inches, altitude 2 inches, draw

(a) A right-angled triangle.

(b) An isosceles triangle.

(c) One or more acute-angled scalene triangles.
(d) One or more obtuse-angled triangles.

Calculate the area of each.

3. Can you show, by cutting from paper, that a right-angled triangle having its base and perpendicular 4 inches and 3 inches, respectively, has the same surface as an acute-angled triangle whose base and altitude are 4 inches and 3 inches, respectively, and an obtuse-angled triangle whose base and altitude are 4 inches and 3 inches, respectively?

Two triangles that have the same area are called equivalent triangles; those having their corresponding sides and angles equal, each to each, are called equal triangles.

4. Construct a triangle whose sides measure 1, 2, and 21 inches, respectively. Construct another triangle having its sides of the same lengths. Are the angles of the second equal to the angles of the first? Are the triangles equal?

5. Draw two triangles each of which has two sides measuring 11⁄2 and 3 inches, respectively, and the included angle 60 degrees. Is the third side of one triangle equal to the third side of the other? Are the remaining angles of the first triangle equal to the remaining angles of the second?

6. With a base 2 inches long, and with angles at the base measuring 50° and 60°, respectively, construct a triangle. Construct a second triangle having a base measuring 2 inches, and angles at the base measuring 60° and 50°, respectively. Are the two triangles equal?

A

B

7. A person wishing to ascertain the length, AB, of a pond, places a pole at a convenient point, C, visible from A and B. The distance BC is measured, and a pole is set up, on a line with B and C, at D, the distance CD being made equal to BC. A pole is also placed at E, on a line with A and C, the distance CE being made equal to AC.

DZ

E

Can you show that the length, AB, of the pond can be ascertained by measuring the distance DE?

SIMILAR TRIANGLES.

1272. Construct a right-angled triangle ABC. Make AB 4 inches, AC 5 inches, BC 3 inches.

At the points n, y, and s, distant from A one, two, and three inches, respectively, erect perpendiculars. Measure A

the length of each.

8. An is one-fourth of AB; ascertain the ratio between mn and CB.

Would the relations as found above, exist if mn were not perpendicular to AB?

How do the angles in the triangle Amn compare with the angles in the triangle ABC?

9. Draw a triangle whose sides measure 4, 5, and 6 inches, respectively. Cut out of paper a triangle whose sides measure 2, 2, and 3 inches, respectively. Place an angle of the small triangle on the corresponding angle of the large triangle, and compare their respective sizes.

How does the area of the small triangle compare with the area of the large triangle?

10. Two angles of a triangle measure about 37° and 53°, respectively. The sides opposite those angles measure 3 inches and 4 inches, respectively. How many degrees does the third angle contain? Calculate the length of the third side.

What will be the dimensions of a similar triangle whose area is one-fourth that of the given triangle? Give the approximate number of degrees in each angle of the small triangle.

11. Draw a line DE, 27 inches long.

At any angle draw

DF. Commencing at D, mark off on DF quarter-inch portions,

Into how many parts is the line ED divided? What fraction of an inch does each part contain?

In locating the points 1, 2, 3, 4, etc., is it necessary to make the divisions inch? Would it be sufficient to use the compasses with the points any convenient distance apart?

12. Divide a line 27 inches long into 5 equal parts.

CALCULATING HEIGHTS AND DISTANCES.

1273. To verify the results obtained by calculation, the pupil should make diagrams, drawing the figures to a convenient scale.

2. A post 6 feet above ground throws a shadow of 71⁄2 feet. How high is a tree whose shadow measures 60 feet?