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Solution. The drawing of the pizza is shown in the figure with the construction lines drawn dashed. Open the compasses and insert the point at 0, and draw an arc AB. Using the same radius, or longer if you wish, insert the point of the compass at A and draw an arc. Then insert the compass point at B, and using the same radius, draw another arc to meet the arc from A atX. The line OX is the angle bisector of angle AOB, and divides the pizza equally. The point X is equidistant from the lines OA and OB.
References: Arc, Bisect, Equidistant, Radius.
26 ANGLE IN A SEMICIRCLE
ANGLE IN A SEMICIRCLE
This is a geometry theorem about a triangle drawn in a semicircle. This theorem is a special case of another theorem given under the entry Angle at the Center and Circumference of a Circle.
In this special case the arc that subtends the angles is the semicircle AB. (See figure a), in which case the angle at the center of the circle is 180° and the angle at the circumference is half of 180°, which is 90°. The theorem is stated as follows:
In figure a, if AS is a diameter of the circle and C is a point on the circumference, then angle ACB is a right angle.
Example. Suppose you are presented with a drawing of a circle and asked to find its diameter.
Solution. This method makes use of the theorem that the angle in a semicircle is a right angle. Tear off a corner of a piece of paper and place it on the circle with the comer of the paper containing the right angle just touching a point on the inside circumference of the circle, as shown in figure b. The sides of the piece of paper will cross over the circle at two points which are labeled A and B in figure b. The dashed line AB is the diameter of the circle.
Example. Joanne loves skating and regularly practices her skills at the Big Apple skating rink, which is in the shape of a circle. One of her activities is to start at a point 5 on the edge of the circle, skate through the center of the rink, and pick up a ball at O (see figure c). Having collected the ball, she carries on in a straight line until she meets the edge of the rink again at the point A. On her way back to her starting point at 5 she touches another point B, which is also on the edge of the rink. If the size of angle ASB = 47°, find the size of angle BAS.
ANGLE IN THE ALTERNATE SEGMENT
Angle ABS = 90° Angle in a semicircle is a right angle
Angle BAS = 43° Angle sum of triangle ABC = 180°
The theorem that the angle in a semicircle is a right angle has a converse theorem, and is stated here:
If there is a right-angled triangle ABC which is right-angled at C (see figure d), and the circumcircle of the triangle is drawn, then line AB is the diameter of the circumcircle.
References: Altitude, Angle Sum of a Triangle, Angles at the Center and Circumference of a Circle, Circumcircle, Converse of a Theorem, Cyclic Quadrilateral, Orthocenter, Semicircle, Subtend.
ANGLE IN THE ALTERNATE SEGMENT
This theorem is a circle geometry theorem. A circle passes through the three vertices of a triangle ABC and at one of these vertices, say C, a tangent is drawn to the circle (see figure a). The chord BC divides the circle into two segments. If one of the segments, say the smaller one, is shaded, then the other segment is referred to as the alternate
28 ANGLE OF DEPRESSION
segment. This theorem states that the angle between the tangent CD and the chord BC is equal to the angle at A in the alternate segment, or
Angle BCD = Angle ВАС
Angle ACE = Angle ABC
Example. Three straight footpaths touch a circular playground at the points A, B, and C, and angle TCA = 65° (see figure b).
(a) What is the size of angle ABC ?
(b) Name another angle the same size as 65°.
Solution, (a) Write
Angle ABC = Angle ACT Angle in the alternate segment
Angle ABC = 65°
(b) Angle ABC is also equal to angle TAC, because of the theorem Angle in the Alternate Segment:
Angle TAC = 65°
References: Angle Sum of a Triangle, Chord, Circle Geometry Theorems, Segment of a Circle, Tangent.
ANGLE OF DEPRESSION
Suppose a surveyor is at the point A on the top of a wall looking horizontally out to sea (see figure). The angle x through which she lowers her gaze to look at a buoy B out at sea is the angle of depression of the buoy from her position A.
Reference: Angle of Elevation.
ANGLE SUM OF A TRIANGLE 29
ANGLE OF ELEVATION
Suppose Darren has climbed to the top of his house and is looking horizontally into the distance (see figure). The angle y through which he raises his eyes to gaze up at the top of a flagpole is the angle of elevation of the top of the flagpole from his position A.
ANGLE OF INCLINATION
This is the angle a certain line makes with another line, or the angle it makes with a plane. Suppose Helen is abseiling down the wall of a building and her rope makes an angle of 33° with the wall (see figure). The angle of inclination of the rope to the wall is 33°. In other words, the rope is inclined at 33° to the wall.