Similar triangles and scale factor

Example 1: find a missing side

Problem $\triangle ABC \sim \triangle DEF$, with $AB = 6$, $BC = 9$ and $DE = 10$. Find EF.

1. AB matches DE, and BC matches EF. Keep the same triangle on top in both fractions: ABC on top, DEF on the bottom. Call EF x.
   $$\frac{6}{10} = \frac{9}{x}$$
2. Cross multiply.
   $$6x = 90$$
3. Divide by 6.
   $$x = 15$$
4. Check: the scale factor from ABC to DEF is $\frac{10}{6} = \frac{5}{3}$, and $9 \cdot \frac{5}{3} = 15$.

Answer $EF = 15$

Example 2: a shadow problem

Problem A student who is 5 feet tall casts a 4-foot shadow. At the same moment, a flagpole casts a 22-foot shadow. How tall is the flagpole?

1. Both the student and the pole stand straight up, so each makes a right angle with the ground. The sun's rays hit both at the same angle. Two equal angles means the two triangles (object, shadow, sun ray) are similar by AA.
2. Height over shadow is the same ratio in both triangles. Call the pole's height h.
   $$\frac{5}{4} = \frac{h}{22}$$
3. Cross multiply.
   $$4h = 110$$
4. Divide by 4.
   $$h = \frac{110}{4} = 27.5$$

Answer 27.5 feet.

Example 3: a triangle inside a triangle

Problem In $\triangle ABC$, point D is on side AB and point E is on side AC, with DE parallel to BC. $AD = 4$, $DB = 6$, and $DE = 5$. Find BC.

1. Because DE is parallel to BC, corresponding angles are equal: $\angle ADE = \angle ABC$. Both triangles also share angle A. So $\triangle ADE \sim \triangle ABC$ by AA.
2. The big triangle's side is all of AB, which is $4 + 6 = 10$. It is not DB.
3. Small triangle on top, big triangle on the bottom. Call BC x.
   $$\frac{4}{10} = \frac{5}{x}$$
4. Cross multiply.
   $$4x = 50$$
5. Divide by 4.
   $$x = \frac{50}{4} = 12.5$$
6. Using DB by mistake gives $\frac{4}{6} = \frac{5}{x}$ and $x = 7.5$, which is shorter than it should be.

Answer $BC = 12.5$

Example 4 (test-hard): scaling area and volume

Problem Two cans are similar cylinders. The small can is 6 cm tall and holds 135 cubic centimeters. The large can is 10 cm tall. (a) How much does the large can hold? (b) The small can's label uses 54 square centimeters of paper. How much paper does the large can's label use?

1. Find the scale factor from small to large.
   $$k = \frac{10}{6} = \frac{5}{3}$$
2. Volume scales by $k^3$.
   $$135 \cdot \left(\frac{5}{3}\right)^{3} = 135 \cdot \frac{125}{27} = 625$$
3. Label area scales by $k^2$.
   $$54 \cdot \left(\frac{5}{3}\right)^{2} = 54 \cdot \frac{25}{9} = 150$$
4. A common wrong answer for (a) is $135 \cdot \frac{5}{3} = 225$. That scales a volume as if it were a length.

Answer (a) 625 cubic centimeters. (b) 150 square centimeters.