PROPOSITION III. THEOREM

40. At a given point in a line only one perpendicular can be erected to that line.

A

B

Let CD be to AB at the point D.

To Prove CD is the only that can be erected to AB at D.

Proof. Suppose a second, as DE, could be erected to AB at D.

NOTE. The points and lines of the above figure, and of all figures given in the first five books of this geometry, are understood to be in the same plane. The term "line" is used in this work for "straight line."

Proof. Suppose them to be unequal and that ABC, when superimposed upon DEF, takes the position GEF.

Then at E there would be two perpendiculars to EF, which contradicts § 40.

Therefore the supposition that the right angles ABC and DEF are unequal is false, and they are equal.

Q.E.D.

42. SCHOLIUM. The right angle is the unit of measure for angles. An angle is generally expressed in terms of the right angle. Thus, AR.A., or B = 14 R.A., etc.

43. DEFINITIONS. In a right-angled triangle the side opposite the right angle is called the hypotenuse.

The other two sides are the legs of

the triangle.

Leg

Hypotenuse

Leg

44. EXERCISE. If two R. A. A have the legs of one equal respectively to the legs of the other, the ▲ are equal in all respects.

45. EXERCISE. A is 40 miles west of B. C is 30 miles north of A, and D is 30 miles south of A.

from D to B?

From C to B is 50 miles.

How far is it

46. EXERCISE. A is m yards north of B. C is n yards west of A, and D is n yards east of B. Prove that the distance from B to C is the same as the distance from A to D.

PROPOSITION IV. THEOREM

47. If a perpendicular is drawn to a line at its middle point,

I. Any point on the perpendicular is equally distant from the extremities of the line.

e II. Any point without the perpendicular is unequally distant from the extremities of the line.

I. Let CD be to AB at its middle point D, and P be any point on CD.

To Prove P equally distant from A and B.

Draw PA and PB.

[It is required to prove PA = PB, for PA and PB measure the distance from P to A and B respectively.]

Proof. The APAD and PBD have

AD = DB (Hypothesis),

<1= <2 (Right Angles),

PD = PD (Common).

The A are equal in all respects by § 30.

:. PA = PB, and P is equally distant from 4 and B.

Q.E.D.

II. Let CD be to AB at its middle point D, and P be any point without CD.

To Prove P unequally distant from A and B.

Draw PA and PB.

[It is required to prove PA and PB unequal.]

Proof. One of these lines, as PA, will intersect the perpendicular CD in some point, as E.

Since PB and PA are unequal, P is unequally distant from A and B.

Q.E.D.

48. COROLLARY I. A perpendicular erected to a line at its middle point contains all points that are equally distant from the extremities of the line.

For, by $ 47, any point on the perpendicular is equally distant from the extremities of the line, and any point without the perpendicular is unequally distant from the extremities of the line. Therefore all points that are equally distant from the extremities of the line must be on the perpendicular.

✓ 49. COROLLARY II. If a line has two of its points each equally distant from the extremities of another line, the first line is perpendicular to the second at its middle point.

Let AB have two of its points m and n each equally distant from the extremities of CD.

To Prove AB L to CD at its middle point.

Proof. Suppose a line were drawn to CD at its middle point.

By § 48 both m and n must be on this perpendicular.

By hypothesis both m and n are on AB.

m

n

So the perpendicular and AB both pass through m and n. By Axiom 13 only one straight line can pass through two given points.

.. AB must coincide with the perpendicular

to CD at its middle point.

50. DEFINITIONS.

Q.E.D.

B

In an isosceles triangle

the angle formed by the two equal sides is called the vertical angle. The side opposite this angle is usually called the base of the triangle.

A

51. EXERCISE. If a perpendicular is erected to the base of an isosceles ▲ at its middle point, it passes through the vertex of the vertical angle.

Suggestion. Use § 48.

52. EXERCISE. If a line is drawn from the vertex of the vertical angle of an isosceles ▲ to the middle point of the base, it is perpendicular to the base.

Suggestion. Use § 49.