Lesson 8 · from §76 of the 1910 text
Miscellaneous Exercises in Proportion
Proportion in geometry
Proportion is the main tool for working with figures of the same shape.
Similar triangles. If one triangle is similar to another, the sides of the first, taken in any order, are proportional to the sides of the second taken in the same order. So if a triangle has sides 6, 8, 10 and a similar one has the side 9 corresponding to 6, the ratio is 6 : 9 = 2 : 3, and the other sides are 12 and 15. The perimeters are in the same ratio.
Parallel lines. A line parallel to one side of a triangle divides the other two sides into four proportional parts. In triangle ABC below, DE is parallel to BC, so \frac{AD}{DB} = \frac{AE}{EC}. It also cuts off a triangle ADE similar to ABC: \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}.
The line joining the middle points of two sides of a triangle is parallel to the third side, so it is half as long as the third side.
Mean proportionals in figures. In a right triangle ABC with the right angle at B, the perpendicular BD to AC is a mean proportional between AD and DC: \frac{AD}{BD} = \frac{BD}{DC}.
In a semicircle ABC, a perpendicular CD from the curve to the diameter AB is a mean proportional between AD and DB.
Example. In the right triangle, if AD = 4 and DC = 9, then BD^2 = 36 and BD = 6.
Exercise 16 uses this layout of stakes on a river bank, where BE is parallel to CD:
Try one:
Quick check
In the right triangle above, AD = 4 and DC = 9. Find BD.
From the history of algebra
Around 600 BC, the Greek Thales of Miletus is said to have measured the height of an Egyptian pyramid by proportion. He waited until his own shadow was as long as he was tall; at that moment the pyramid's shadow (measured from the center of its base) was as long as the pyramid was high. Two centuries later Eudoxus worked out a careful theory of proportion that handles even lengths with no common measure, and Euclid made it Book V of his Elements. Merchants for centuries used the "rule of three," finding a fourth proportional to three given numbers, much as in this chapter.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Give numbers only, or expressions where letters appear.
If a : b = 4 : x, and a : b = 4 : \dfrac1x, find x.
If \dfrac ab = \dfrac9y, and \dfrac ab = \dfrac{1}{\frac1y}, find y.
If p : r = 7 : 6, and q : r = 3 : 5, find the ratio p : q (as a fraction).
If p : q = 4 : a, and q : r = a : 7, find the ratio p : r (as a fraction).
The sides of a triangle are 8, 10, and 12. The side 12 is divided into the ratio of the other two sides. Find the two parts.
The perimeter of a triangle is 63. Two sides are 18 and 24 and the other side is divided in the ratio of these two. Find the two parts of the third side.
A flagstaff casts a shadow 12 yards long; at the same time a man 5 feet 10 inches tall casts a shadow 35 inches long. How high is the pole?
The sides of a triangle are 10, 15, and 20 respectively. In a similar triangle the side corresponding to 10 is 12. Find the other sides. Compare the ratio of the two corresponding sides with the ratio of the perimeters.
The sides of a triangle are 9, 10, and 17. The perimeter of a similar triangle is 108. Find the sides of the second triangle.
In triangle ABC (the first figure above), DE is parallel to BC.
Draw a triangle and letter the vertices F, G, and H respectively. Draw RK parallel to FG, R being on side HF and K on side HG. Then:
Two sides of a triangle are 30 centimeters and 24 centimeters respectively. The line joining their middle points is 12 centimeters long. Find the third side of the triangle.
The sides of a triangle are 10, 12, and 16 centimeters respectively. Find the lengths of the lines connecting the middle points of its sides.
In the right-angled triangle ABC (the second figure above), line BD is perpendicular to AC. Then BD is a mean proportional between AD and DC.
In the semicircle ABC (the third figure above), line CD is perpendicular to AB. Then CD is a mean proportional between AD and DB.
The distance AB between two points on opposite banks of a river was wanted. Stakes were set at E, B, D, and C, so that BE was parallel to CD, and so that ABC and AED were straight lines (the last figure above). The measured values of DC, CB, and BE were 480 feet, 160 feet, and 420 feet respectively. What was the computed value of BA?
The perimeters of two similar triangles are 45 and 135 respectively. One side of the first is 11 and a second side is 19. Find the sides of the second triangle.
Two men start at the same time and travel in opposite directions. The ratio of their rates is 2 : 3. In 5 hours they are 100 miles apart. Find the rate of each.
A, B, and C ate equally of a stock of provisions which A and B furnished. The values of what A and B contributed were in the ratio of 7 to 8 respectively. C paid $30 for what he ate. How should A and B have divided the money?
A clock provided with hands to indicate the minute, the hour, and the day of the month showed correct time at 4 P.M. on February 21, 1900. The clock gained 10 minutes daily. What was the correct time when the clock indicated 4 P.M. on the 28th of the next month? (Give the hour and the whole number of minutes; the answer is in the morning of March 28.)