Draw a model of $\sqrt{12}$ using perfect squares.
Answer (2)
Express 12 in terms of perfect squares using the Pythagorean theorem.
Recognize that 12 = 16 − 4 , where 16 and 4 are perfect squares.
Construct a right triangle with hypotenuse 4 and one leg 2, making the other leg 12 .
The final model is a right triangle with sides 2, 12 , and 4, representing the relationship between the perfect squares 4 and 16: $\boxed{\sqrt{12}}.
Explanation
Problem Analysis Let's analyze the problem. We need to create a model of 12 using perfect squares. This means we need to find a geometric representation where 12 is a side of a shape, and the other sides or areas are related to perfect squares (numbers like 1, 4, 9, 16, etc.).
Using Pythagorean Theorem One way to approach this is to use the Pythagorean theorem, which relates the sides of a right triangle: a 2 + b 2 = c 2 , where a and b are the lengths of the legs, and c is the length of the hypotenuse. We want to find a right triangle where one of the sides is 12 , and the squares of the other sides are perfect squares.
Expressing 12 as Difference of Perfect Squares We can express 12 as the difference of two perfect squares: 16 − 4 = 12 . This suggests that we can create a right triangle where the hypotenuse is 4 (since 4 2 = 16 ) and one leg is 2 (since 2 2 = 4 ). Then, the other leg will be 12 . Let a = 12 , b = 2 , and c = 4 . Then, ( 12 ) 2 + 2 2 = 4 2 , which simplifies to 12 + 4 = 16 , which is true.
Drawing the Right Triangle So, we can draw a right triangle with hypotenuse of length 4, one leg of length 2, and the other leg of length 12 . The squares of the sides are 16, 4, and 12 respectively.
Alternative Representation Alternatively, we can express 12 as 2 3 . We can represent 3 as a leg of a right triangle with hypotenuse 2 and one leg 1, since 1 2 + ( 3 ) 2 = 2 2 . Then, we can double the length of 3 to get 12 .
Final Model The model consists of a right triangle with hypotenuse of length 4, one leg of length 2, and the other leg of length 12 . The squares of the sides are 16, 4, and 12 respectively, representing the perfect squares.
Final Answer The final answer is a right triangle with a hypotenuse of 4 and one leg of 2. The other leg has a length of 12 .
Examples
Understanding square roots and their geometric representation using perfect squares is crucial in various fields. For example, in construction, when calculating the length of a diagonal support beam in a rectangular structure, you often use the Pythagorean theorem. If you know the lengths of the two sides forming the right angle, you can find the length of the diagonal (hypotenuse) which involves square roots. Similarly, in navigation, calculating distances 'as the crow flies' often involves square roots and the Pythagorean theorem, especially when dealing with coordinates on a map.
To model 12 using perfect squares, express 12 as the difference between perfect squares, such as 16 − 4 . Construct a right triangle with one leg of length 2 , the hypotenuse 4 , and the other leg representing 12 . This representation illustrates the relationship between the square root and perfect squares visually.
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