How to Find Unknown Lengths (Sides) of a Similar Triangle using Euclid's Theorem
Similarity is a concept Euclidean geometry uses to describe the commonalities between and consequent properties of two individual shapes. Similar triangles have two things in common: two angles and shape. Since the sum of a triangle's angles must always equal 180 degrees, having two shared angles means that the degree of the third angle must also be the same between the pair of triangles. Having the same shape, means that the sides of the triangle have the same proportions. When you know that two triangles are similar, you can make assumptions about the shapes and calculate the degree of unknown angles and the length of unknown sides.
Instructions
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1
Draw the two triangles and fill in as much information as you can. For example, triangle ABC has two known angles and two known sides: angles ABC and CAB are both 60 degrees, and side A and side B are both 5 inches. Triangle DEF has two known angles and one known sides: angle DEF and FDE are 60 degrees and side E is 10 inches.
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2
Fill in the missing information for the first triangle, ABC. Subtract the sum of the known angle from 180 to find the degree of the unknown angle, 60 degrees.
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3
Use the properties of the triangle to dictate the length of the unknown side. In this case, three 60-degree angle identifies ABC as an equilateral triangle, in which case side C is also 5 inches. If the triangle is a right-triangle use the Pythagorean theorem, a^2 + b^2 =c^2, to find the length of the unknown side.
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4
Use knowledge of triangles to fill in information, For example, since side E is 10 inches and it is known that triangle DEF is an equilateral triangle, side D and F are also 10 inches.
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5
Fill in the rest of the missing information for the second triangle, DEF, using the information gleamed from ABC. Divide the value of the known side of DEF with the value of the corresponding side in ABC; divide side E with side B to find the multiplier, 2. Multiply the value of each side of ABC with the multiplier to find the corresponding value of each side in triangle DEF. Side E, 10, divided by side B, 5, gives a multiplier of 2, so the sides of triangle DEF will be twice the length of ABC.
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Tips & Warnings
Perimeters can also be calculated using the multiplier found in Step 5. The perimeter of ABC is 15 inches. Multiply the value by 2 to reach the perimeter of DEF, 30 inches. You can prove that the perimeter is in fact 30 inches by adding the value of each side.
References
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