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Let the distance to work be x kilometers. Therefore the total distance traveled is 2x kilometers. On the outward journey the time taken is Ti hours and for the return journey the time is F2 hours. Write
T\ = — and T2 = — The formula used is time = distance speed.
Now write Total time = Ij + T2
total distance traveled
Formula for average speed
total time taken
x 4 x 3
= — x —[-------x- The lowest common denominator is 120
30 4 40 3
Simplifying each fraction
Adding fractions with equal denominators
ALTERNATE ANGLES 17
Substituting distance and time into average speed formula
Multiplying by the reciprocal of
x is a common factor of numerator and denominator
Canceling the x’s The average speed is 34.3 km h_1, to one decimal place.
References: Average Speed, Canceling, Equivalent Fractions, Rational Expression, Reciprocal. ALTERNATE ANGLES
Figure a shows two parallel lines indicated by arrows. The line cutting across them is called a transversal. A pair of angles such as a and b that are on alternate sides of the transversal and lie between the parallel lines are equal in size and are called alternate angles.
Alternate angles form a shape like a reversed letter z, as shown in figure b.
Angle a = angle &
The pair of alternate angles shown in figure c forms a letter z:
Angle c = angle d
Average speed =-------------
5 F (7*/120)
2x x 120
~T x —
18 ALTERNATE ANGLES
Example. Figure d shows a concrete pillar set in the seabed as a support for a bridge. If a laser beam focused on the foot of the pillar makes an angle of 47° with the sea level, find the angle, marked x, that the laser beam makes with the line of the seabed. The seabed is parallel to the sea level.
x = 47° Alternate angles are equal
The laser beam makes an angle of 47° with the seabed.
There are two more geometry theorems that relate to two parallel lines and their transversal. They are as follows:
1. Corresponding angles are equal in size.
2. Cointerior angles are supplementary, which means they add together to equal 180°.
With reference to figure e, a pair of angles that are on the same side of the transversal and occupy a similar position are equal in size and are called corresponding angles. Angles a and b are corresponding angles, and so are angles c and d:
Angle a = angle &
Angle c = angled
ALTERNATE ANGLES 19
Figure f shows two more pairs of corresponding angles that are equal in size:
Angle e = angle /
Angle g = angle h
Example. A ladder leans against a vertical wall, and the angle the ladder makes with the horizontal ground is 65°. Find the angle the ladder makes with the top of the wall (figure g).
x = 65° Corresponding angles are equal The ladder makes an angle of 65° with the top of the wall.
Cointerior angles are pairs of angles that lie between two parallel lines and are situated on the same side of the transversal:
The sum of cointerior angles = 180°
In figure h, angles a and b are cointerior, and so are c and d:
Angle a + angle b = 180°
Angle c + angle d — 180°
20 ALTERNATE ANGLES
Example. Figure i shows a chimney on the roof of a house. One side of the chimney makes an angle of 110° with the roof. Find the size of the angle the other side of the chimney makes with the roof, marked x in the figure.
x + 110° = 180° Sum of cointerior angles = 180°
x =70° Subtracting 110° from both sides of the equation
The chimney makes an angle of 70° with the roof.
Example. In figure j, the two angles of a triangle are 60° and 70°, and the arrows indicate parallel lines. Find the sizes of the angles x, y, and z.
y + 60° + 70° y + 130°
x = 50° z = 60°
Sum of the angles of a triangle = 180°
Subtracting 130° from both sides of the equation Alternate angles are equal Corresponding angles are equal
From the results of the above calculations it can be seen that the sum of the three angles of the triangle is the same total as the sum of the three angles on the straight line, and that the sum is 180°.
References: Angle Sum of a Triangle, Geometry Theorems.
This word is used in two different ways. The altitude of an object is the distance of the object above the surface of the earth, and is often called its vertical height. In geometry, altitude takes on a slightly different meaning, and refers to the altitude of a polygon or a polyhedron. In this context, the term altitude is explained under the entry Base (geometry).
References: Base (geometry), Concurrent, Polygon, Polyhedron.
Amplitude is a feature of periodic curves, like the sine or the cosine curves. The amplitude of the sine curve is the greatest distance of a point on the curve from the x-axis, and is indicated by a in the figure. For the sine curve y = sinx, the amplitude is a = 1. For the curve y = 2 cos x, the amplitude is a = 2.