This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown leg with given hypotenuse and one leg. For a right triangle with hypotenuse c=10 units and leg a=6 units, the other leg b is b² = c² - a² = 100 - 36 = 64, so b = √64 = 8 units. In this specific problem, with hypotenuse 10 units and one leg 6 units, the other leg is √(10² - 6²) = √(100 - 36) = √64 = 8 units. The correct setup identifies the hypotenuse, subtracts the squared leg, and takes the square root, matching choice B. A common error is adding instead of subtracting, like 100 + 36 = 136, or not taking square root to choose 64. To solve: (1) identify the right triangle, (2) label hypotenuse 10 units, one leg 6 units, other b, (3) note hypotenuse and one leg known, (4) set up b² = 10² - 6², (5) calculate 100 - 36 = 64, b = √64 = 8 units, (6) verify it's reasonable as 8 is between 6 and 10. Common mistakes include mislabeling hypotenuse or errors in squaring like 6²=36 but 10²=100 correct.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or 3D problems like space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, finding the hypotenuse by plugging in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or finding a leg by rearranging to a² = c² - b², such as if c = 13 and b = 5, then a² = 169 - 25 = 144, so a = 12; in real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base = 9 ft and ladder = 15 ft, and solve 9² + h² = 15² to get h = 12 ft. In this specific problem, the right triangle has a hypotenuse of 13 m and one leg of 5 m, so we find the other leg using a² = 13² - 5², giving 169 - 25 = 144, so a = √144 = 12 m. The correct setup involves identifying c = 13 m as the hypotenuse and b = 5 m as one leg, then rearranging to a² = c² - b², calculating the squares, subtracting, and taking the square root to get 12 m, which matches choice B. A common error might be subtracting incorrectly, like 13 - 5 = 8, or forgetting to take the square root and choosing 144 m, or confusing which side is the hypotenuse and using the formula wrong. To solve these problems, follow these steps: (1) identify the right triangle with a 90° angle, (2) label the sides with legs a and b forming the right angle and hypotenuse c opposite it as the longest side, (3) identify the knowns (hypotenuse and one leg) and unknown (other leg), (4) set up a² = c² - b², (5) calculate by squaring the knowns, subtracting, and taking the square root, (6) verify it makes sense, like 12 m being between 5 m and 13 m. Common mistakes include using a + b = c without squaring, misidentifying the hypotenuse, arithmetic errors in squaring or subtracting, or taking a negative square root.

This question tests applying the Pythagorean theorem a² + b² = c² to find the diagonal of a rectangle, which forms a right triangle. For a rectangle 6 inches wide (a=6 in) and 8 inches tall (b=8 in), the diagonal d is the hypotenuse: d² = 6² + 8² = 36 + 64 = 100, so d = √100 = 10 in. In this specific poster problem, the diagonal across 6 in by 8 in is found using d = √(6² + 8²) = √(36 + 64) = √100 = 10 in. The correct setup treats the width and height as legs of a right triangle, adds their squares for d², and takes the square root, matching choice B. A common error is multiplying like 6 × 8 = 48, or adding without squaring to get 14, or choosing 100 without square root. To solve: (1) identify the right triangle formed by the diagonal, (2) label legs as 6 in and 8 in, diagonal as d (hypotenuse), (3) note both legs known, hypotenuse unknown, (4) set up 6² + 8² = d², (5) calculate 36 + 64 = 100, d = √100 = 10 in, (6) verify it's reasonable as 10 in is longer than both sides. Common mistakes include confusing diagonal with perimeter or arithmetic errors like 8² = 64 but adding to 100 incorrectly.

This problem tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or even 3D space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, to find the hypotenuse, plug in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or to find a leg, rearrange to a² = c² - b², such as if c=13 and b=5, then a² = 169 - 25 = 144, so a=12. In real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base=9 ft and ladder=15 ft, and solve with 9² + h² = 15² to get h=12 ft. In this specific rectangle problem, the rug is 6 ft wide and 8 ft long, so the diagonal is found using d² = 6² + 8² = 36 + 64 = 100, giving d = √100 = 10 ft. The correct setup involves recognizing the diagonal forms a right triangle with the sides 6 ft and 8 ft as legs, applying a² + b² = d², and calculating to get 10 ft. Common errors include multiplying like 6*8=48, mistaking 6² as 12, forgetting the square root and saying 100 ft, or treating it as a non-right triangle. To solve: (1) identify the right triangle formed by the diagonal, (2) label legs a=6 ft and b=8 ft, hypotenuse d unknown, (3) note two legs known, hypotenuse unknown, (4) set up a² + b² = d², (5) calculate 36 + 64 = 100, √100=10, (6) verify 10 ft is reasonable as it's longer than both sides. Common mistakes: a + b = d like 6+8=14, wrong hypotenuse, arithmetic errors in addition or squaring, or negative length.

This question tests applying the Pythagorean theorem in 3D to find the space diagonal of a rectangular prism. For a box with dimensions 3 cm, 4 cm, 12 cm, the space diagonal d uses d² = 3² + 4² + 12² = 9 + 16 + 144 = 169, so d = √169 = 13 cm, extending the theorem to three dimensions. In this specific prism problem, the space diagonal from one corner to the opposite is √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13 cm. The correct setup squares all three dimensions, adds them, and takes the square root, matching choice A. A common error is using only two dimensions, like √(3² + 4²)=5, or confusing with surface diagonal to get 12.65 or similar. To solve: (1) identify the 3D right triangle path, (2) label dimensions as three perpendicular legs, (3) note all known, diagonal unknown, (4) set up d² = 3² + 4² + 12², (5) calculate 9 + 16 + 144 = 169, d = √169 = 13 cm, (6) verify it's reasonable as 13 cm is longer than the longest side 12 cm. Common mistakes include omitting one dimension or arithmetic like 12²=124.

This question tests applying the Pythagorean theorem $a^2 + b^2 = c^2$ to find the unknown hypotenuse in a right triangle with given legs. For a right triangle with legs a=6 cm and b=8 cm, and hypotenuse c (the longest side opposite the 90° angle), plug the legs into the formula: $6^2 + 8^2 = 36 + 64 = 100 = c^2$, so c = $\sqrt{100}$ = 10 cm. In this specific problem, the right triangle has legs of 6 cm and 8 cm, so the hypotenuse is found using c = $\sqrt{(6^2 + 8^2)} = \sqrt{(36 + 64)} = \sqrt{100}$ = 10 cm. The correct setup involves identifying the legs as a and b, squaring them, adding, and taking the square root for c, which matches choice A. A common error is adding without squaring, like 6 + 8 = 14, or forgetting the square root and choosing 100 cm. To solve: (1) identify the right triangle, (2) label legs as 6 cm and 8 cm, hypotenuse as c, (3) note both legs known, hypotenuse unknown, (4) set up $6^2 + 8^2 = c^2$, (5) calculate 36 + 64 = 100, c = $\sqrt{100}$ = 10 cm, (6) verify it's reasonable as 10 cm is longer than both legs. Common mistakes include confusing legs with hypotenuse or arithmetic errors like $6^2 = 36$ but $8^2 = 68$ incorrectly.

This problem tests applying the Pythagorean theorem $a^2 + b^2 = c^2$ to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or even 3D space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using $a^2 + b^2 = c^2$; for example, to find the hypotenuse, plug in the legs like $6^2 + 8^2 = 36 + 64 = 100 = c^2$, so $c = \sqrt{100} = 10$, or to find a leg, rearrange to $a^2 = c^2 - b^2$, such as if c=13 and b=5, then $a^2 = 169 - 25 = 144$, so a=12. In real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base=9 ft and ladder=15 ft, and solve with $9^2 + h^2 = 15^2$ to get h=12 ft. In this specific problem, the right triangle has hypotenuse 13 units and one leg 5 units, so the other leg is found using $a^2 = 13^2 - 5^2 = 169 - 25 = 144$, giving a = $\sqrt{144}$ = 12 units. The correct setup involves identifying the hypotenuse as c=13 and leg b=5, rearranging the formula to $a^2 = c^2 - b^2$, and calculating the square root to get 12 units. Common errors include subtracting without squaring like 13-5=8, mistaking 5² as 10, forgetting the square root and saying 144 units, or confusing the hypotenuse and using it as a leg. To solve: (1) identify the right triangle with the 90° angle implied, (2) label legs a (unknown) and b=5, hypotenuse c=13 as the longest, (3) note hypotenuse and one leg known, other leg unknown, (4) set up $a^2 = c^2 - b^2$, (5) calculate 169 - 25 = 144, $\sqrt{144}=12$, (6) verify $5^2 + 12^2 = 25 + 144 = 169 = 13^2$. Common mistakes: using a + b = c forgetting squares, wrong side as hypotenuse like treating 5 as c, arithmetic errors in subtraction or square root, or negative length.

This problem tests applying the Pythagorean theorem $a^2 + b^2 = c^2$ to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or even 3D space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using $a^2 + b^2 = c^2$; for example, to find the hypotenuse, plug in the legs like $6^2 + 8^2 = 36 + 64 = 100 = c^2$, so $c = \sqrt{100} = 10$, or to find a leg, rearrange to $a^2 = c^2 - b^2$, such as if c=13 and b=5, then $a^2 = 169 - 25 = 144$, so a=12. In real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base=9 ft and ladder=15 ft, and solve with $9^2 + h^2 = 15^2$ to get h=12 ft. In this specific problem, the right triangle has legs 8 in and 15 in, so the hypotenuse is found using $c^2 = 8^2 + 15^2 = 64 + 225 = 289$, giving $c = \sqrt{289} = 17$ in. The correct setup involves identifying legs a=8 and b=15, applying $a^2 + b^2 = c^2$, and calculating the square root to get 17 in. Common errors include adding without squaring like 8+15=23, mistaking 8² as 16, forgetting the square root and saying 289 in, or confusing hypotenuse with a leg. To solve: (1) identify the right triangle with 90° implied, (2) label legs a=8 in and b=15 in, hypotenuse c unknown, (3) note two legs known, hypotenuse unknown, (4) set up $a^2 + b^2 = c^2$, (5) calculate 64 + 225 = 289, $\sqrt{289}=17$, (6) verify 17 in is reasonable as longest side. Common mistakes: $a + b = c$ (23 in), wrong hypotenuse, arithmetic errors in addition or root, negative values.

This question tests applying the Pythagorean theorem a² + b² = c² to find an unknown leg in a right triangle with given hypotenuse and one leg. For a right triangle with hypotenuse c=13 m and leg b=5 m, rearrange to a² = c² - b²: a² = 169 - 25 = 144, so a = √144 = 12 m. In this specific problem, with hypotenuse 13 m and one leg 5 m, the other leg is found using a = √(13² - 5²) = √(169 - 25) = √144 = 12 m. The correct setup involves identifying the hypotenuse as the longest side, subtracting the squared leg from the squared hypotenuse, and taking the square root, matching choice B. A common error is subtracting incorrectly or using addition like 13 + 5 = 18, or not taking the square root and choosing 144 m. To solve: (1) identify the right triangle, (2) label hypotenuse as 13 m, one leg as 5 m, other as a, (3) note hypotenuse and one leg known, other leg unknown, (4) set up a² = 13² - 5², (5) calculate 169 - 25 = 144, a = √144 = 12 m, (6) verify it's reasonable as 12 m is between 5 m and 13 m. Common mistakes include treating the given leg as hypotenuse or arithmetic errors like 13² = 179.

This problem tests applying the Pythagorean theorem a² + b² = c² to find an unknown side in right triangles, such as 2D problems like ladders or diagonals, or even 3D space diagonals. For a right triangle with legs a and b, and hypotenuse c (the longest side opposite the 90° angle), if two sides are known, the third can be found using a² + b² = c²; for example, to find the hypotenuse, plug in the legs like 6² + 8² = 36 + 64 = 100 = c², so c = √100 = 10, or to find a leg, rearrange to a² = c² - b², such as if c=13 and b=5, then a² = 169 - 25 = 144, so a=12. In real-world scenarios, identify the right triangle, like a ladder against a wall forming a right angle, assign values such as base=9 ft and ladder=15 ft, and solve with 9² + h² = 15² to get h=12 ft. In this specific coordinate problem, points A(0,0) and B(9,12) form a right triangle with legs 9 and 12 (differences in x and y), so distance d = √(9² + 12²) = √(81 + 144) = √225 = 15. The correct setup involves recognizing the distance formula as Pythagorean theorem with Δx=9 and Δy=12 as legs, calculating to get 15. Common errors include adding coordinates like 0+9=9, mistaking differences, forgetting square root and saying 225, or not squaring. To solve: (1) identify right triangle from axes, (2) label legs Δx=9 and Δy=12, hypotenuse d, (3) note legs known, d unknown, (4) set up d² = Δx² + Δy², (5) calculate 81 + 144 = 225, √225=15, (6) verify 15 is reasonable. Common mistakes: a + b = d like 9+12=21, arithmetic errors, wrong differences, negative values.