Use similar triangles and scale factors

• Establish triangle similarity from matching angle information.
• Pair corresponding sides in a consistent order before writing proportions.
• Use one scale factor to solve and verify all available side relationships.

Build the idea

Worked example

1. Step 1

   The side D E corresponds to A B. Dividing fifteen by five gives an enlargement factor of three from the first triangle to the second.

   $$k=\frac{15}{5}=3$$

2. Step 2

   The side E F corresponds to B C, so multiply the original length seven by the same factor three.

   $$EF=3\cdot7=21$$

3. Step 3

   Compare the two corresponding ratios. Both enlarged-over-original quotients equal three, confirming consistent correspondence.

   $$\frac{15}{5}=\frac{21}{7}=3$$

• Pairing sides because they look similarly oriented in a rotated drawing.

  Use equal-angle markings or the written vertex order; orientation can change under rotation or reflection.

• Flipping only one ratio in a proportion.

  Choose enlarged over original or original over enlarged and use that direction for every corresponding pair.

Two similar triangles have corresponding sides eight and twelve. If another side of the smaller triangle is ten, find its partner in the larger triangle.

This reviewed reading order covers the lesson explanation, equations, example, and checks without claiming that a video exists.

1. Part 1

   Lesson overview

   Similar triangles have equal corresponding angles and proportional corresponding sides. A single scale factor converts every length in one triangle to its partner. Similarity preserves shape while allowing size to change. Equal corresponding angles fix the shape, and every corresponding length is multiplied by the same positive number. Establish triangle similarity from matching angle information.

2. Part 2

   Visual model and equations

   Vertex order matters because it records correspondence. Writing triangle A B C similar to triangle D E F means A matches D, B matches E, and C matches F. Place a three-four-five right triangle beside an enlarged triangle whose shortest side is six. Matching angle arcs show correspondence, and every enlarged side is twice its partner. D E over A B equals E F over B C equals D F over A C, all equal to scale factor k. Angle A equals angle D, and angle B equals angle E.

   • D E over A B equals E F over B C equals D F over A C, all equal to scale factor k.
   • Angle A equals angle D, and angle B equals angle E.
3. Part 3

   Worked example

   Triangle A B C is similar to triangle D E F. A B is five, B C is seven, D E is fifteen, and E F is unknown. Find E F. Use the declared vertex order to pair A B with D E and B C with E F, find the enlargement factor, and apply it to the unknown side. The side D E corresponds to A B. Dividing fifteen by five gives an enlargement factor of three from the first triangle to the second. The side E F corresponds to B C, so multiply the original length seven by the same factor three. Compare the two corresponding ratios. Both enlarged-over-original quotients equal three, confirming consistent correspondence. The corresponding side E F has length twenty-one. The computed side gives the same positive scale factor as the known side pair, and the vertex order pairs the endpoints consistently.

   • The scale factor equals fifteen divided by five, which is three.
   • E F equals three times seven, which is twenty-one.
   • Fifteen over five equals twenty-one over seven, and both equal three.
4. Part 4

   Checks, practice, and scope

   A proportion is trustworthy only when every numerator and denominator use the same direction between figures. Checking a second side pair confirms that the chosen scale factor is consistent. Pairing sides because they look similarly oriented in a rotated drawing. Use equal-angle markings or the written vertex order; orientation can change under rotation or reflection. Flipping only one ratio in a proportion. Choose enlarged over original or original over enlarged and use that direction for every corresponding pair. Two similar triangles have corresponding sides eight and twelve. If another side of the smaller triangle is ten, find its partner in the larger triangle. The enlargement factor is twelve divided by eight, or three halves. The corresponding larger side is fifteen. Equal corresponding angles establish the shared triangle shape. Vertex order or angle marks determine which sides correspond. One scale factor must work for every corresponding length pair. The lesson assumes similarity has been established and does not prove every similarity criterion in full. Scale factors compare lengths; areas scale by the square and volumes by the cube of the length factor.

Scope and limitations

• The lesson assumes similarity has been established and does not prove every similarity criterion in full.
• Scale factors compare lengths; areas scale by the square and volumes by the cube of the length factor.