Example Question - triangle calculation
Here are examples of questions we've helped users solve.
Calculating height of a rectangle using Pythagorean theorem
1. Identificar las longitudes dadas: base (b = 35 cm) y diagonal (d = 37 cm). 2. Aplicar el teorema de Pitágoras: a² + b² = d², donde a es la altura buscada. 3. Sustituir los valores conocidos en la ecuación: a² + 35² = 37². 4. Calcular 35²: 35² = 1225. 5. Calcular 37²: 37² = 1369. 6. Reemplazar los cuadrados en la ecuación: a² + 1225 = 1369. 7. Despejar a²: a² = 1369 - 1225. 8. Realizar la resta: a² = 144. 9. Hallar la raíz cuadrada de 144: a = √144. 10. Calcular √144: a = 12. 11. Concluir que la altura (a) del rectángulo es 12 cm.
Solving Triangles using the Pythagorean Theorem
Para resolver este problema, vamos a calcular el cuadrado de cada lado de los triángulos dados y verificar si cumplen con el teorema de Pitágoras, cuyo enunciado es \(a^2 + b^2 = c^2\), donde \(c\) es la hipotenusa y \(a\) y \(b\) son los catetos de un triángulo rectángulo. Triángulo 1: - \(3^2 + 4^2 = 9 + 16 = 25\) - \(5^2 = 25\) Triángulo 2: - \(5^2 + 12^2 = 25 + 144 = 169\) - \(13^2 = 169\) Triángulo 3: - \(8^2 + 15^2 = 64 + 225 = 289\) - \(17^2 = 289\) Todos los triángulos cumplen con el teorema de Pitágoras; por lo tanto, todos son triángulos rectángulos.
Calculation of Hypotenuse Using Pythagorean Theorem
La imagen muestra tres triángulos, y la instrucción pide calcular el cuadrado de los tres lados de cada triángulo y comprobar que se cumple el Teorema de Pitágoras, que establece que en un triángulo rectángulo, el cuadrado de la hipotenusa (el lado opuesto al ángulo recto) es igual a la suma de los cuadrados de los otros dos lados. Vamos a resolver cada triángulo por separado: 1) Para el primer triángulo: - Lado a = 13 cm - Lado b = 84 cm Aplicamos el Teorema de Pitágoras: \(a^2 + b^2 = c^2\) \(13^2 + 84^2 = c^2\) \(169 + 7056 = c^2\) \(7225 = c^2\) \(c = \sqrt{7225}\) \(c = 85 cm\) 2) Para el segundo triángulo: - Lado a = 50 cm - Lado b = 120 cm Aplicamos el Teorema de Pitágoras: \(a^2 + b^2 = c^2\) \(50^2 + 120^2 = c^2\) \(2500 + 14400 = c^2\) \(16900 = c^2\) \(c = \sqrt{16900}\) \(c = 130 cm\) 3) Para el tercer triángulo: - Lado a = 45 cm - Lado b = 28 cm Aplicamos el Teorema de Pitágoras: \(a^2 + b^2 = c^2\) \(45^2 + 28^2 = c^2\) \(2025 + 784 = c^2\) \(2809 = c^2\) \(c = \sqrt{2809}\) \(c = 53 cm\) De este modo, hemos calculado la hipotenusa de los tres triángulos y verificado que se cumple el Teorema de Pitágoras en cada caso.
Calculating the Missing Side Length in a Right-Angled Triangle
The image shows a right-angled triangle with one of the sides adjacent to the right angle labeled "24 cm" and the hypotenuse labeled "26 cm." To find the length of the other side adjacent to the right angle (often referred to as the opposite side or the height in the context of triangles), we can use the Pythagorean theorem. 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. The theorem is usually written as: \[ c^2 = a^2 + b^2 \] where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides. Given that \( c = 26 \) cm and \( b = 24 \) cm, we can solve for \( a \): \[ 26^2 = a^2 + 24^2 \] \[ 676 = a^2 + 576 \] \[ a^2 = 676 - 576 \] \[ a^2 = 100 \] \[ a = \sqrt{100} \] \[ a = 10 \] Therefore, the length of the other side adjacent to the right angle (the opposite side) is 10 cm.
Finding the Length of a Missing Leg in a Right Triangle
The image shows a right triangle with the lengths of two sides provided: one side is 75 cm, and the hypotenuse is 72 cm. The question asks for the length of the missing leg (we will call this b). To find the length of the missing leg, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The Pythagorean theorem is written as follows: c^2 = a^2 + b^2 In this case, we will rearrange the equation to solve for b: b^2 = c^2 - a^2 We know c is 75 cm and a is 72 cm, so let's plug these numbers into the equation: b^2 = 75^2 - 72^2 b^2 = 5625 - 5184 b^2 = 441 Now, to find the length of side b, we take the square root of both sides: b = √441 b = 21 cm Therefore, the length of the missing leg b is 21 centimeters.
Solving for the Lengths of Legs in an Isosceles Right-Angled Triangle
This image shows a right-angled triangle with one of the angles being \(45^\circ\), which makes it an isosceles right-angled triangle (since the other non-right angle must also be \(45^\circ\)). The hypotenuse of this triangle is given as 5 units. In an isosceles right-angled triangle, the lengths of the two legs (sides opposite the \(45^\circ\) angles) are equal. If we let one of the legs be \(x\), we can use the Pythagorean theorem to solve for \(x\): \[x^2 + x^2 = 5^2\] \[2x^2 = 25\] \[x^2 = \frac{25}{2}\] \[x = \sqrt{\frac{25}{2}}\] \[x = \frac{5}{\sqrt{2}}\] Multiplying the numerator and denominator by \(\sqrt{2}\) to rationalize the denominator, we get: \[x = \frac{5\sqrt{2}}{2}\] So the lengths of the two legs are each \(\frac{5\sqrt{2}}{2}\) units.
Calculation of Leg Length in 45-45-90 Triangle
This image depicts a right triangle, where one of the angles is 45 degrees and the hypotenuse opposite this angle measures 5 units. Since this is a 45-45-90 triangle, the two legs are congruent. In a 45-45-90 right triangle, the legs are each \( \frac{1}{\sqrt{2}} \) (which is the same as \( \sqrt{2}/2 \)) times the length of the hypotenuse. Let's call the length of each leg \( x \). Then: \[ x = \frac{1}{\sqrt{2}} \times 5 = \frac{5}{\sqrt{2}} \] However, it's often preferred to rationalize the denominator, so we multiply the numerator and denominator by \( \sqrt{2} \): \[ x = \frac{5\sqrt{2}}{\sqrt{2}\times\sqrt{2}} = \frac{5\sqrt{2}}{2} \] Therefore, the length of each leg of the triangle is \( \frac{5\sqrt{2}}{2} \) units.
Solving for the Length of Legs in a 45-45-90 Triangle
The image shows a right-angle triangle with a 45-degree angle. Since the angles in a triangle add up to 180 degrees and we already have a right angle (90 degrees) and one 45-degree angle, the other angle must also be 45 degrees. This makes the triangle a 45-45-90 triangle, which is a form of an isosceles right triangle. In a 45-45-90 triangle, the lengths of the legs (the two shorter sides) are equal, and the length of the hypotenuse (the longest side, opposite the right angle) is √2 times the length of a leg. The figure gives the hypotenuse as length 3. Therefore, to find the length of each leg (let's call it 'L'), we can use the proportion that L is to 3 (the hypotenuse) as 1 is to √2: L / 3 = 1 / √2 Multiplying both sides by 3 to solve for L: L = 3 / √2 To rationalize the denominator: L = (3 / √2) * (√2 / √2) L = (3√2) / 2 So, the length of each leg of the triangle is (3√2) / 2 units.
Solving a Plane Geometry Problem using Pythagorean Theorem
这个问题是一个利用勾股定理来解决的平面几何问题。首先，我们有一个直角三角形ABC，其中∠ACB是直角。 已知两条边： AB = 2； AC = 2√3。 我们需要计算第三边BC的长度。 根据勾股定理，直角三角形的斜边（即此处的BC）的平方等于两条直角边的平方和，即： BC² = AB² + AC²。 带入已知数值： BC² = 2² + (2√3)² BC² = 4 + 4*3 BC² = 4 + 12 BC² = 16。 现在我们要找BC的长度，所以我们取平方根： BC = √16 BC = 4。 因此，BC的长度是4。在提供的选项中没有直接为4的选项。所以可能图片问题选择题部分有笔误或看错数字。在标准数学问题条件下，BC的正确答案应该是4。
Calculating the Length of a Hypotenuse
The Pythagorean theorem states that for a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The theorem can be written as: c² = a² + b² In the provided right-angled triangle, side a is 9 ft and side b is 12 ft. You are asked to find the length of the hypotenuse (c). Using the Pythagorean theorem: c² = 9² + 12² c² = 81 + 144 c² = 225 To find the length of c, take the square root of both sides: c = √225 c = 15 ft Therefore, the length of the hypotenuse is 15 feet.
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