8. One angle of a triangle is 48° 20′ 10′′ and another is 76° 42' 56". Find the third angle.

9. One angle of a triangle is 96° 42′ 43′′ and another is 83° 17′ 16′′. Find the third angle.

10. In ▲ ABC, ▲ A=34°, and ≤ B=62°, and side a = § in. Construct the triangle. Which side is the longest? Which is the shortest?

11. Can you construct a triangle ABC with ▲ A=127°, LB 64°, and AB=2 in.? If you cannot, give a reason.

81. The perpendicular bisector of a line.

1. Take a point P. How many different straight lines can be drawn to pass through P?

2. Take two points M and N. How many different straight lines can be drawn to pass through both M and N?

3. If you wish to draw a certain line and must first find some points on it, how many points must you find in order to be able to draw it?

The answers to these questions make clear the following Principle. Through two points only one straight line can be drawn.

This principle may be stated in the form :

Two points determine a straight line. This means that one straight line and only one can be drawn through the two points.

If a line is divided into two equal parts, it is said to be bisected.

82. To construct the perpendicular bisector of a given line segment. We wish to construct the perpendicular bisector of the segment AB, that is, to construct a line which is perpendicular to AB, and which bisects AB.

How many points on the perpendicular bisector must we know in order to be able to draw it?

Now find

If P is a point on the perpendicular bisector, how does the distance from P to A compare with the distance from P to B? With the compasses find such a point. another such point, Q. If you then join the points P and Q, you will have the perpendicular bisector of AB.

The construction may be made as follows: With A and B as centers and the same radius construct arcs meeting at P and Q. Draw the line PQ, which intersects AB in a point M. Then PQ is the perpendicular bisector and M is the mid-point of AB.

Exercise 81

1. Draw a line segment and bisect it.

PX

A

M B

Qx

FIG. 56.

2. With ruler and compasses make the following figures.

3. With ruler and compasses make the following figures.

4. In a given circle draw a chord. Draw the perpendicular bisector of the chord. Does the perpendicular bisector

of the chord pass through the center of the circle? Draw another chord and its perpendicular bisector. Where do the two perpendicular bisectors meet?

5. If you do not know where the center of a given circle is, how can you find it?

6. If you were given part of the rim of a wheel, as in Figure 61, how could you find the length of its radius?

Principle. The perpendicular bisector

of a chord passes through the center of the circle.

FIG. 61.

83. To erect a perpendicular to a line at a given point on the line.

We wish to erect a perpendicular to the line AB at the point P.

With P as center construct an arc

cutting AB at C and D.

With C as center and radius greater

than one-half CD, construct an arc.

P

D B

FIG. 62.

With D as center and the same

radius construct an arc cutting the last arc at Q. Draw PQ. Then PQ is perpendicular to AB at P.

Then PQ is the perpendicular to the line AB from the

point P.

85. The distance from a point to a line. If you were in a field and wished to take the shortest path to a straight road AB, you would follow the perpendicular from the point to the line AB.

By the distance from a point to a line we mean the length of the perpendicular from the point to the line.

86. To bisect a given angle.

An angle is bisected by a line which divides it into two equal angles.

It is required to bisect ABC.
With B as center draw an arc

cutting AB at R and BC at S.

B

'R

FIG. 64.

With R and S as centers and the same radius draw arcs cutting each other at P. Draw the line PB.

Then PB bisects the ZABC.

Exercise 82

1. Along what line would you measure the distance from a point on the blackboard to the upper edge of the blackboard?

2. A straight railroad runs through a town. What is meant by the distance from a point in the town to the railroad track? Draw a figure to illustrate your answer.

8. Construct the perpendicular bisector of a given line. Test the accuracy of your construction by measuring the angles.

4. Draw a triangle ABC. Construct the perpendicular bisector of each side. If the construction is accurate, these bisectors will meet in a point. Call this point O. Using the compasses, compare the distances OA, OB, and OC. Can you now draw a circle passing through A, B, and C?

5. Take three points not in a straight line. Construct a circle passing through these three points.

HINT. Join the three points, thus forming a triangle, and use the method of the preceding exercise.

6. Construct an angle of 45°.

HINT. Construct a right angle and bisect it.

7. What angle, if bisected, will give an angle of 22° 30'? Construct an angle of 22° 30'.

8. Draw a triangle and construct the bisector of each of its angles. If the constructions are accurate, these bisectors will meet in a point.

9. Erect a perpendicular to the line AB from the external point C.

10. Define the altitude of a triangle.

11. Draw an acute triangle. Construct its three altitudes. If the constructions are accurate, the three altitudes will meet in a point.

12. Draw an obtuse triangle, and construct its three altitudes. The same test applies as in the preceding exercise.

13. Draw a right triangle. Construct its three altitudes. Do they meet in a point? Where?

14. Construct these eight-pointed stars.

HINT.

Draw two perpendicular diameters and bisect the angles between them to obtain the vertices.