Example Question - solving for side
Here are examples of questions we've helped users solve.
Solving for the Length of a Side in a Right-Angled Triangle
The image shows a right-angled triangle, which means we can use the Pythagorean theorem to solve for the missing side, labeled as "x." The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In the given triangle, we have one side that is 9 cm and another side that is 12 cm. Assuming that the 12 cm side is the hypotenuse (since it's opposite what is presumably the right angle), we can compute the length of side "x" using the following equation: \( c^2 = a^2 + b^2 \) Here, \( c \) represents the hypotenuse, while \( a \) and \( b \) represent the other two sides. Rearranging the equation to solve for \( a \), we get: \( a^2 = c^2 - b^2 \) Let's substitute the given values: \( x^2 = 12^2 - 9^2 \) \( x^2 = 144 - 81 \) \( x^2 = 63 \) Taking the square root of both sides to solve for \( x \), we find: \( x = \sqrt{63} \) The square root of 63 can be simplified further by factoring out perfect squares. \( 63 = 9 \times 7 \), and since \( 9 \) is a perfect square, we get: \( x = \sqrt{9 \times 7} \) \( x = \sqrt{9} \times \sqrt{7} \) \( x = 3\sqrt{7} \) Therefore, the length of side \( x \) is \( 3\sqrt{7} \) cm, which cannot be simplified further without a decimal approximation. If a decimal approximation is needed, \( \sqrt{7} \approx 2.645 \), and multiplying this by 3 gives: \( x \approx 3 \times 2.645 \) \( x \approx 7.935 \) So the side "x" is approximately 7.935 cm long.
Solving for the Length of a Side in a Right-Angled Triangle Using Trigonometry
The image shows a right-angled triangle with sides labeled, and we need to solve for side \( x \). This can be done using trigonometry. The triangle has one side of length 77, which is the opposite side to the angle marked as 68 degrees, and side \( x \) is the hypotenuse. To find side \( x \), we can use the sine (sin) function, which is defined as the ratio of the length of the opposite side over the hypotenuse in a right-angled triangle: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] Here, \( \theta \) is 68 degrees, the opposite side is 77, and the hypotenuse is \( x \). Plugging these into the equation: \[ \sin(68^\circ) = \frac{77}{x} \] We now want to solve for \( x \): \[ x = \frac{77}{\sin(68^\circ)} \] Using a calculator: \[ x \approx \frac{77}{0.927} \] \[ x \approx \frac{77}{0.927} \] \[ x \approx 83.1 \] Thus, the length of side \( x \) is approximately 83.1, to the nearest tenth.
Solving for the Length of a Side in a Right-Angled Triangle
The image shows a right-angled triangle with sides labeled. The hypotenuse is labeled as 73, one leg is labeled as 48, and the other leg, which is opposite the right angle, is labeled with the variable N. To solve for N, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The Pythagorean theorem formula is as follows: c² = a² + b² In this case: c = 73 (the hypotenuse), a = 48 (one of the other sides), b = N (the side we're trying to find). Plugging the numbers we know into the formula gives us: 73² = 48² + N² Now calculate each term: 73² = 5329, 48² = 2304, And putting them into the equation: 5329 = 2304 + N² Next, subtract 2304 from both sides to solve for N²: 5329 - 2304 = N² 3025 = N² Finally, take the square root of both sides to find N: N = √3025 N = 55 Therefore, the value of N is 55.
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