Since the sides are in ratio 5:12:13 and the shortest side is 15, the scale factor is $$\frac{15}{5} = 3$$. The three sides are $$5 \times 3 = 15$$, $$12 \times 3 = 36$$, and $$13 \times 3 = 39$$. The area is $$\frac{1}{2} \times 15 \times 36 = \frac{540}{2} = 270$$ square units. Choice A uses the original 5:12 ratio: $$\frac{1}{2} \times 5 \times 36 = 90$$. Choice B uses $$\frac{1}{2} \times 15 \times 24 = 180$$. Choice D uses $$\frac{1}{2} \times 15 \times 26 = 195$$.

The area of the triangle is $$\frac{1}{2}x^2 = 18$$, so $$x^2 = 36$$ and $$x = 6$$. In an isosceles right triangle, if the legs have length 6, the hypotenuse has length $$6\sqrt{2}$$. Choice A gives the leg length instead of hypotenuse. Choice C incorrectly adds 6 + 3. Choice D uses $$x = 9$$ instead of $$x = 6$$, giving $$9\sqrt{2}$$.

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.

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 $$\sqrt{4 + 3} = \sqrt{7}$$.

When you see a question about distances on a square field, you're dealing with the Pythagorean theorem and comparing direct versus indirect paths.

First, let's find the distance from home plate to second base. Since a baseball diamond is a square with 90-foot sides, home plate and second base are diagonally opposite corners. Using the Pythagorean theorem: $$d = \sqrt{90^2 + 90^2} = \sqrt{2 \cdot 90^2} = 90\sqrt{2}$$ feet.

The indirect route (home to first base, then first base to second base) covers two sides of the square: $$90 + 90 = 180$$ feet.

The difference between these distances is: $$180 - 90\sqrt{2} = 90(2 - \sqrt{2})$$ feet.

Looking at the wrong answers: Choice A gives $$90(\sqrt{2} - 1)$$, which would result from incorrectly calculating $$90\sqrt{2} - 90$$ instead of $$180 - 90\sqrt{2}$$. Choice B is $$90\sqrt{2}$$, which is just the diagonal distance itself, not the difference between routes. Choice C suggests 45 feet, which might come from thinking the difference is simply half a side length.

The correct answer is D: $$90(2 - \sqrt{2})$$ feet.

Strategy tip: In geometry problems involving squares and diagonals, always identify whether you need the Pythagorean theorem for the diagonal, and be careful about what the question is actually asking for—sometimes it's the distance itself, sometimes it's a comparison between distances.

In a square with side length $$s$$, the diagonal equals $$s\sqrt{2}$$. Given diagonal = $$8\sqrt{2}$$, we have $$s\sqrt{2} = 8\sqrt{2}$$, so $$s = 8$$. The perimeter is $$4s = 4(8) = 32$$ centimeters. Choice A gives the side length instead of perimeter. Choice C squares the side length: $$8^2 = 64$$. Choice D incorrectly uses $$2 \cdot 8\sqrt{2} = 16\sqrt{2}$$.

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 $$25 - 15 = 10$$.

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.