Idiot\'s Guides Basic Math and Pre-Algebra

238 Part 3: The Shape of the World

If you know triangles are similar, you know angles are congruent and sides are in proportion.

Suppose nRST,nABC, RS = 12 feet, BC = 6 feet and ST = 18 feet, and you need to find the

length of AB. Use the fact that in similar triangles, corresponding sides are in proportion:

RS

AB

ST

BC

5. Fill in the known lengths.^1218

x 6

. Cross-multiply and solve. 18x = 72, and x = 4. The

length of AB is 4 feet.

Indirect Measurement with Similar Triangles

Right triangles are often helpful in finding areas of irregular figures and in finding

measurements indirectly, as you’ll see a little later. Triangles of many different shapes can be

used for indirect measurement, if you use what you know about similarity.

Remember the two triangle relationships, congruence and similarity? Either one can help you

find a measurement you can’t take directly, but in cases where you can’t measure directly, you

may not be able to create an exact copy to use congruent figures. Similarity is easier because

you can make a smaller copy of the same shape and then use similar triangles to find the

measurement of the larger version.

Similar polygons are the same shape but not necessarily the same size. Because their sides are in

proportion, however, you can use a smaller version of the figure to calculate the measurements of

a larger one.

If you need to know how wide the river is from point W on one bank to point A on the other

bank, you can’t walk on water to take the measurement. But you can measure off 100 meters

along one bank and mark point T, then go another 20 meters and mark point R. Create 'RET

to be similar to 'WAT by making side RE parallel to side WA, and ’WTA = ’RTE. (Okay, you

probably don’t walk around with a protractor in your pocket, but it’s possible to do it.) Measure

RE and set up the proportion WA

RE

WT

TR

. You can plug in what you know, and solve for WA.

A

W 100

12

TR 20

E

?