Common Mistakes
Overview
Learn to identify and avoid the most common errors students make with transformations and the Pythagorean theorem. Recognizing these mistakes will help you succeed on tests.
Practice Problems
Question 1: A student uses a^2 + b^2 = c^2 and gets 3^2 + 4^2 = 9 + 16 = 25, so c = 25. What's wrong?
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Answer: Forgot to take the square root
c^2 = 25 means c = sqrt(25) = 5, not c = 25. Always take the square root at the end.
Question 2: A student reflects (4, -3) over the x-axis and gets (-4, 3). Find the error.
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Answer: Changed the wrong coordinate
Reflection over x-axis changes the y-coordinate sign: (4, -3) becomes (4, 3), not (-4, 3).
Question 3: A triangle has sides 5, 6, and 8. A student concludes it's a right triangle because 5 + 6 > 8. Why is this incorrect?
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Answer: Mixed up the triangle inequality with Pythagorean theorem
To check for a right triangle, test if a^2 + b^2 = c^2: 25 + 36 = 61, but 64 is not 61. Not a right triangle.
Question 4: A student says that after a 180-degree rotation, a shape is always congruent but also always in the same position. What's the error?
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Answer: The position changes (unless the center of rotation is the center of the shape)
Rotations are rigid motions that preserve size and shape but change position and orientation.
Question 5: A student finds the hypotenuse as sqrt(5^2 - 3^2) = sqrt(16) = 4. What went wrong?
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Answer: Used subtraction instead of addition for finding hypotenuse
To find hypotenuse: c^2 = a^2 + b^2, not c^2 = a^2 - b^2. Subtraction is for finding a leg when given hypotenuse.
Question 6: A student translates (2, 5) left 3 and up 4, getting (5, 9). Identify the mistake.
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Answer: Added to x instead of subtracting for "left"
Left means subtract from x: (2-3, 5+4) = (-1, 9), not (5, 9).
Question 7: A student dilates a triangle by scale factor 2 and says the area doubles. Is this correct?
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Answer: No - area quadruples (scale factor squared)
When linear dimensions are multiplied by k, area is multiplied by k^2. Scale factor 2 means area increases by 2^2 = 4.
Question 8: A student tries to use the Pythagorean theorem on a triangle with sides 4, 5, 6 where 4 and 5 are not the legs. What should they check first?
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Answer: Check if it's actually a right triangle first
The Pythagorean theorem only works for right triangles. Test: 4^2 + 5^2 = 41, but 6^2 = 36. Not equal, so not a right triangle.
Question 9: When rotating 90 degrees counterclockwise, a student changes (3, 4) to (3, -4). Find the correct answer.
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Answer: The correct answer is (-4, 3)
90 degrees counterclockwise: (x, y) becomes (-y, x). So (3, 4) becomes (-4, 3).
Question 10: A student says dilation is a rigid transformation because the shape stays the same. Explain the error.
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Answer: Dilation is NOT a rigid transformation because it changes size
Rigid transformations (translations, rotations, reflections) preserve distance. Dilation changes distance, so the shape is similar but not congruent.