Overview: What Are Similar Triangles?
Two triangles are similar if they each have two angles that have the same measurement, and the lengths of the corresponding sides are proportionate. Therefore, one triangle can be made to measure the same as the other one by multiplying each side by the same factor. For example, given two triangles ∆ABC and ∆DEF, angles A and D measure the same, angles B and E measure the same, and angles C and F measure the same. The angles that measure the same are shown with the same symbol on each triangle.
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What Are Proportionate Sides?
If the lengths of corresponding sides are in the same ratio to each other, then they are proportionate. For example, in the above similar triangles, the length of DF is 4, the length of DE is 5, and the length of EF is 6. The length of AC is 8, the length of AB is 10, and the length of BC is 12. The ratio of DF/AC is 4/8, simplified as 1/2, the ratio of DE/AB is 5/10, simplified to 1/2, and the ratio of EF/BC is 6/12, simplified to 1/2.
Using Algebra to Find Unknown Measures
Suppose all that was known were that two triangles were similar, the measurements of three sides of one triangle, and the measurements of two sides of the other similar triangle. In this example, the measurement of DE is 3, DF is 5, and EF is 7. The measurement of AB is 6 and BC is 10. What is the measurement of AC? In this example, AB/DE =AC/EF. Substituting the measurements of sides, 6/3 =x/7, where x represents the measurement of AC. So 3x = 7(6), or 3x = 42. Dividing both sides by 3, x =14. The measurement of AC is 14.
Are Special Triangles Similar?
The isosceles right triangle, with measurements of 900, 450, and 450, is one type of special triangle. The lengths of its sides are determined by the Pythagorean theorem. Both sides that make up the right angle have the same length, a ,and the hypotenuse equals a√2. Similarly, a right triangle with measurements 300, 600, and 900, will have a shorter leg of b, a longer leg of b√3, and a hypotenuse of 2b. Any isosceles right triangle will be similar to any other isosceles right triangle, because they have angle measures in common and side lengths in proportion, and any 300-600-900 right triangle will be similar to any other, but the two types of right triangles are not similar because they only have one angle, the right angle, in common.
Can Other Polygons Be Similar?
In order for polygons to be similar to each other, both the proportionality requirement and the equal angle requirement must be met. For example, the definition of a square is that all four sides are equal and that all angles are right angles. All squares are similar to one another, because their angles have a common measure, and their sides are in proportion.
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