APPLICATIONS OF PROPORTIONS

Similar figures have exactly the same shape but not necessarily the same size. Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if and only if the lengths of corresponding sides are proportional and all pairs of corresponding angles have equal measures.

AB/DE  =  BC/EF  =  AC/DF

m∠A  =  m∠D

m∠B  =  m∠E

m∠C  =  m∠F

When stating that two figures are similar, use the symbol ∼. For the triangles above, you can write ΔABC ∼ ΔDEF. Make sure corresponding vertices are in the same order. It would be incorrect to write ΔABC ∼ ΔEFD.

You can use proportions to find missing lengths in similar figures.

Cross Products Property

Words :

In a proportion, cross products are equal.

Numbers :

Algebra :

Finding Missing Measures in Similar Figures

Example 1 :

In the diagram shown below ΔRST ∼ ΔBCD. Find the length of BC.

Solution :

R corresponds to B, S corresponds to C, and T corresponds to D.

RT/BD  =  RS/BC

5/12  =  8/x

Use Cross Product Property. 

5(x)  =  12(8)

5x  =  96

Because x is multiplied by 5, divide each side by 5 to undo the multiplication.

5x/5  =  96/5

x  =  19.2

The length of BC is 19.2 ft.

Example 2 :

In the diagram shown below FGHJKL ∼ MNPQRS. Find the length of QR.

Solution :

GH/NP  =  JK/QR

4/6  =  2/x

Use Cross Product Property.

4(x)  =  6(2)

4x  =  12

Because x is multiplied by 4, divide each side by 4 to undo the multiplication.

4x/4  =  12/4

x  =  3

The length of QR is 3 cm.

Indirect Measurement

You can solve a proportion involving similar triangles to find a length that is not easily measured. This method of measurement is called indirect measurement. If two objects form right angles with the ground, you can apply indirect measurement using their shadows.

Measurement Application

Example 3 :

A totem pole casts a shadow 45 feet long at the same time that a 6-foot-tall man casts a shadow that is 3 feet long. Write and solve a proportion to find the height of the totem pole.

Solution :

Both the man and the totem pole form right angles with the ground, and their shadows are cast at the same angle. You can form two similar right triangles.

6/x  =  3/45

Use Cross Product Property. 

6(45)  =  x(3)

270  =  3x

Because x is multiplied by 3, divide each side by 3 to undo the multiplication.

270/3  =  3x/3

90  =  x

The totem pole is 90 feet tall.

Scale Factor

If every dimension of a figure is multiplied by the same number, the result is a similar figure. The multiplier is called a scale factor.

Changing Dimensions

Example 4 :

Every dimension of a 2-by-4-inch rectangle is multiplied by 1.5 to form a similar rectangle. How is the ratio of the perimeters related to the ratio of corresponding sides? How is the ratio of the areas related to the ratio of corresponding sides?

Solution :

The ratio of the perimeters is equal to the ratio of corresponding sides.

The ratio of the areas is the square of the ratio of corresponding sides.

Example 5 :

Every dimension of a cylinder with radius 4 cm and height 6 cm is multiplied by ½ to form a similar cylinder. How is the ratio of the volumes related to the ratio of corresponding dimensions?

Solution :

The ratio of the volumes is the cube of the ratio of corresponding dimensions.

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