Apply Pythagorean Theorem to Problems - 8th Grade Math

A right triangle has legs of length 6 cm and 8 cm. What is the length of the hypotenuse?
Answer: 10 cm 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.
Refer to the figure below. A right triangle has legs of length $x$ and $x+7$ , and a hypotenuse of length $13$ . What is the value of $x$ ?
Answer: $5$ Using $x^2 + (x+7)^2 = 13^2$ : $x^2 + x^2 + 14x + 49 = 169$ , giving $2x^2 + 14x - 120 = 0$ , or $x^2 + 7x - 60 = 0$ . Factoring: $(x-5)(x+12)=0$ , so $x = 5$ . Legs are $5$ and $12$ , hypotenuse $13$ . Choice A gives legs $3$ and $10$ : $\sqrt{109} \ne 13$ . Choice C gives legs $6$ and $13$ : not correct. Choice D is the other leg's length, not $x$ .
A 15-foot ladder leans against a vertical wall. The bottom of the ladder is 9 feet from the wall on level ground. About how high up the wall does the ladder reach?
Answer: 12 ft 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 ladder problem, the ladder is 15 ft (hypotenuse) and base is 9 ft from the wall, so the height is found using h² = 15² - 9² = 225 - 81 = 144, giving h = √144 = 12 ft. The correct setup involves recognizing the right triangle formed by the wall, ground, and ladder, with c=15 ft and b=9 ft, rearranging to h² = c² - b², and calculating to get 12 ft. Common errors include adding instead of subtracting like 225 + 81, mistaking 9² as 18, forgetting the square root and saying 144 ft, or not identifying the right triangle setup. To solve: (1) identify the right triangle with 90° at the wall base, (2) label legs base=9 ft and height h, hypotenuse ladder=15 ft, (3) note hypotenuse and one leg known, other leg unknown, (4) set up h² = c² - b², (5) calculate 225 - 81 = 144, √144=12, (6) verify 12 ft is reasonable as 9² + 12² = 81 + 144 = 225 = 15². Common mistakes: using a + b = c like 9+15=24, confusing hypotenuse, arithmetic errors in squares or subtraction, or negative values.
A right triangle has a hypotenuse of 17 units and one leg of 8 units. What is the length of the other leg?
Answer: 15 units 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 17 units and one leg of 8 units, so the other leg is found using a² = 17² - 8² = 289 - 64 = 225, a = √225 = 15 units. The correct setup identifies c = 17 and b = 8, rearranges to a² = c² - b², calculates squares, subtracts, and takes square root to get 15 units, matching choice B. A common error might be adding squares instead of subtracting, like 289 + 64 = 353, or not squaring and doing 17 - 8 = 9, or choosing 289 without square root. To solve these problems, follow these steps: (1) identify the right triangle with 90° angle, (2) label legs and hypotenuse correctly, (3) identify knowns and unknown, (4) set up a² = c² - b², (5) calculate accordingly, (6) verify 15 is between 8 and 17. Common mistakes: forgetting squares, wrong hypotenuse, arithmetic slips, negative length.
A rectangular storage box measures 3 in. by 4 in. by 12 in. What is the length of the space diagonal (the distance from one corner of the box to the opposite corner)?
Answer: 13 in. 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 3D problem, the rectangular box measures 3 in., 4 in., 12 in., so the space diagonal is d = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169 = 13 in. The correct setup extends Pythagorean to 3D by finding the diagonal of the base (√(3² + 4²) = 5) then with height √(5² + 12²) = √(25 + 144) = √169 = 13 in., or directly √(a² + b² + c²), matching choice A. A common error might be just adding dimensions 3 + 4 + 12 = 19, or using only two sides like √(3² + 4²) = 5 and stopping, or confusing with surface diagonal. To solve these problems, follow these steps: (1) identify 3D right triangle path through space, (2) label dimensions as three perpendicular legs, (3) known all three, unknown space diagonal, (4) set up d = √(a² + b² + c²), (5) calculate squares, add, square root, (6) verify 13 in. longer than 12 in. Common mistakes: not using all dimensions, arithmetic errors, forgetting it's 3D extension, negative root.
A rectangular garden has a diagonal walkway that measures 25 feet. If the garden is 15 feet wide, what is the length of the garden?
Answer: 20 feet Using the Pythagorean theorem: $a^2 + b^2 = c^2$ . Here, width = 15 feet, diagonal = 25 feet, and we need to find length. So $15^2 + length^2 = 25^2$ , which gives us $225 + length^2 = 625$ . Therefore $length^2 = 400$ , so $length = 20$ feet. Choice B incorrectly adds 15 + 25 - 11. Choice C uses $15 + 25 = 40$ . Choice D uses $25 - 15 = 10$ .
A movie screen is shaped like a rectangle that is 9 feet tall and 12 feet wide. What is the length of the diagonal across the screen?
Answer: 15 ft 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 screen is 9 ft tall and 12 ft wide, so the diagonal is found using d² = 9² + 12² = 81 + 144 = 225, giving d = √225 = 15 ft. The correct setup involves recognizing the diagonal forms a right triangle with legs 9 ft and 12 ft, applying a² + b² = d², and calculating to get 15 ft. Common errors include adding without squaring like 9+12=21, mistaking 9² as 18, forgetting the square root and saying 225 ft, or not setting up the right triangle. To solve: (1) identify the right triangle via the diagonal, (2) label legs a=9 ft and b=12 ft, hypotenuse d unknown, (3) note two legs known, hypotenuse unknown, (4) set up a² + b² = d², (5) calculate 81 + 144 = 225, √225=15, (6) verify 15 ft is reasonable as longer than sides. Common mistakes: a + b = d (21 ft), arithmetic errors like wrong squaring, confusing legs, or negative length.
Refer to the figure below. A wire is stretched from the top of a $24$ -foot pole to a point on the ground $18$ feet from the base of the pole. A second wire runs from the same point on the ground to the top of a $10$ -foot pole located $8$ feet from that point (on the opposite side). What is the total length of both wires (to the nearest tenth)?
Answer: $42.8$ ft First wire: $\sqrt{24^2 + 18^2} = \sqrt{576+324} = \sqrt{900} = 30$ ft. Second wire: $\sqrt{10^2 + 8^2} = \sqrt{100+64} = \sqrt{164} \approx 12.8$ ft. Total: $30 + 12.8 \approx 42.8$ ft. Choice A is only the first wire. Choice C is only the second wire. Choice D adds the given ground/pole lengths incorrectly.
A right circular cone has a height of 12 inches and a base radius of 5 inches. What is the slant height of the cone?
Answer: 13 inches When you encounter a cone problem asking for slant height, you're dealing with a 3D geometry situation that creates a right triangle. The key insight is recognizing that the height, radius, and slant height of a cone form a right triangle where the slant height is the hypotenuse. To find the slant height, use the Pythagorean theorem: $a^2 + b^2 = c^2$ . Here, the height (12 inches) and base radius (5 inches) are the two legs, and the slant height is the hypotenuse. So: $12^2 + 5^2 = s^2$ , where $s$ is the slant height. Calculating: $144 + 25 = s^2$ , so $s^2 = 169$ . Taking the square root: $s = 13$ inches. Looking at the wrong answers: Choice A (7 inches) represents the difference between height and radius (12 - 5 = 7), which is a common mistake when students subtract instead of using the Pythagorean theorem. Choice B ( $\sqrt{119}$ ) comes from incorrectly adding the squares: some students mistakenly calculate $\sqrt{12^2 - 5^2} = \sqrt{144 - 25} = \sqrt{119}$ , using subtraction instead of addition in the Pythagorean theorem. Choice C (17 inches) results from simply adding the height and radius (12 + 5 = 17), treating this as a linear addition problem rather than a right triangle. Remember: whenever you see height, radius, and slant height in cone problems, immediately think "right triangle" and reach for the Pythagorean theorem. The slant height is always the hypotenuse.
A rectangular prism (box) has dimensions 3 cm by 4 cm by 12 cm. What is the length of the space diagonal from one corner of the box to the opposite corner?
Answer: 13 cm 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.
A ramp forms a right triangle with the ground. The ramp is 10 ft long (the hypotenuse) and reaches a platform that is 6 ft high. How far is the bottom of the ramp from the platform (horizontal distance)?
Answer: 8 ft 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 ramp problem, the ramp is 10 ft (hypotenuse) and height is 6 ft (one leg), so the horizontal distance is b = √(10² - 6²) = √(100 - 36) = √64 = 8 ft. The correct setup identifies c = 10 ft and a = 6 ft, rearranges to b² = c² - a², calculates, and takes square root to get 8 ft, matching choice B. A common error might be adding squares instead, like 100 + 36 = 136, or not squaring and doing 10 - 6 = 4, or choosing 100 without root. To solve these problems, follow these steps: (1) identify right triangle from ramp and ground, (2) label legs as height and base, hypotenuse as ramp, (3) knowns: hypotenuse and height, unknown base, (4) set up b² = c² - a², (5) calculate, (6) verify 8 ft makes sense with 10 ft ramp. Common mistakes: no squares, wrong subtraction, arithmetic errors, negative value.
A soccer player runs 9 m east and then 12 m north, making a right angle. How far is the player from the starting point?
Answer: 15 m This question tests applying the Pythagorean theorem a² + b² = c² to find the straight-line distance after perpendicular paths. For paths of 9 m east (a=9 m) and 12 m north (b=12 m), the distance c is c² = 9² + 12² = 81 + 144 = 225, so c = √225 = 15 m. In this specific soccer player problem, running 9 m east and 12 m north, the distance from start is √(9² + 12²) = √(81 + 144) = √225 = 15 m. The correct setup treats the paths as legs of a right triangle, adds their squares for c², and takes the square root, matching choice B. A common error is adding the distances like 9 + 12 = 21, or multiplying to 108, or choosing 144 without square root. To solve: (1) identify the right angle from directions, (2) label legs 9 m and 12 m, hypotenuse as distance c, (3) note both legs known, (4) set up 9² + 12² = c², (5) calculate 81 + 144 = 225, c = √225 = 15 m, (6) verify it's reasonable as 15 m is shorter than sum but straight-line. Common mistakes include not recognizing the right triangle or errors in squaring like 9²=91.
A student walks 3 blocks east and then 4 blocks north, making a right angle turn. About how many blocks is the straight-line distance from the starting point to the ending point?
Answer: 5 blocks 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, walking 3 blocks east and 4 blocks north forms a right triangle, so the straight-line distance is the hypotenuse d = √(3² + 4²) = √(9 + 16) = √25 = 5 blocks. The correct setup treats the directions as legs a = 3 and b = 4, with d as c, squares them, adds, and takes square root to get 5 blocks, matching choice A. A common error might be just adding 3 + 4 = 7 without squaring, or subtracting to get 1, or not recognizing the right angle turn implies a right triangle. To solve these problems, follow these steps: (1) identify the right triangle from perpendicular paths, (2) label legs as east and north distances, hypotenuse as straight line, (3) identify known legs and unknown hypotenuse, (4) set up a² + b² = c², (5) calculate squares, add, square root, (6) verify 5 is longer than 4 but shorter than path walked. Common mistakes: a + b = c, arithmetic like 3² = 6, forgetting square root, wrong setup.
A right triangle has legs 8 inches and 15 inches. What is the length of the hypotenuse?
Answer: 17 in 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.
In a right triangle, the hypotenuse is 13 units and one leg is 5 units. What is the length of the other leg?
Answer: 12 units 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.
In a right triangle, the hypotenuse is 13 m and one leg is 5 m. What is the length of the other leg?
Answer: 12 m 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.
Refer to the triangle below. Triangle ABC is a right triangle with the right angle at C. If AC = 12 and BC = 9, what is the length of the altitude from C to the hypotenuse AB?
Answer: $\frac{36}{5}$ units First find the hypotenuse: $AB = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15$ . The area can be calculated two ways: $\frac{1}{2} \cdot 12 \cdot 9 = 54$ and $\frac{1}{2} \cdot 15 \cdot h = \frac{15h}{2}$ . Setting equal: $54 = \frac{15h}{2}$ , so $h = \frac{108}{15} = \frac{36}{5}$ . Choice A stops at $\frac{108}{15}$ without simplifying. Choice C incorrectly uses $\frac{108}{5}$ by using 5 instead of 15. Choice D uses incorrect formula setup.
A right triangle has legs of length 7 units and 24 units. What is the length of the hypotenuse?
Answer: 25 units This question tests applying the Pythagorean theorem $a^2 + b^2 = c^2$ to find the hypotenuse with given legs. For legs a=7 units and b=24 units, $c^2 = 7^2 + 24^2 = 49 + 576 = 625$ , so $c = \sqrt{625} = 25$ units. In this specific problem, with legs 7 units and 24 units, the hypotenuse is $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$ units. The correct setup squares the legs, adds, and takes the square root for c, matching choice A. A common error is adding without squaring like 7 + 24 = 31, or squaring incorrectly to get 49 + 576 = 625 but choosing 625 without root. To solve: (1) identify the right triangle, (2) label legs 7 and 24, hypotenuse c, (3) note legs known, (4) set up $7^2 + 24^2 = c^2$ , (5) calculate 49 + 576 = 625, c = $\sqrt{625} = 25$ units, (6) verify it's reasonable as 25 is longer than 24. Common mistakes include misidentifying hypotenuse or arithmetic like $24^2=574$ .
In the coordinate plane, point A is at $(3, 4)$ and point B is at $(7, 1)$ . What is the distance between points A and B?
Answer: 5 units Use the distance formula, which is based on the Pythagorean theorem: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ . Here: $d = \sqrt{(7-3)^2 + (1-4)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ . Choice B adds the horizontal and vertical distances: $4 + 3 = 7$ . Choice C forgets to take the square root of 25. Choice D incorrectly uses $\sqrt{4 + 3} = \sqrt{7}$ .
A rectangular classroom rug is 6 feet wide and 8 feet long. What is the length of the rug's diagonal (corner to corner)?
Answer: 10 ft 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.

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