Example Question - triangle sides
Here are examples of questions we've helped users solve.
Calculating the Hypotenuse of a Right Triangle
Given \( a = 3 \) and \( b = 4 \) \[ c = \sqrt{a^2 + b^2} \] \[ c = \sqrt{3^2 + 4^2} \] \[ c = \sqrt{9 + 16} \] \[ c = \sqrt{25} \] \[ c = 5 \]
Finding the Value of tan(pi/2 - alpha) in Trigonometry
La pregunta en la imagen es acerca de encontrar el valor de \( tan\left( \frac{\pi}{2} - \alpha \right) \) basado en el triángulo mostrado. Vamos a resolverlo paso por paso. Dado que tenemos un triángulo rectángulo, podemos utilizar la relación entre los lados para encontrar el valor de las funciones trigonométricas del ángulo α. Vemos que el triángulo tiene: - Un lado opuesto al ángulo α de longitud 4 - Un lado adyacente al ángulo α de longitud \( \sqrt{20} \) - La hipotenusa opuesta al ángulo recto de longitud 2 (la letra C está al lado de este valor, sugiriendo que señala el ángulo recto) Sin embargo, hay un error aquí; normalmente, la hipotenusa sería el lado más largo del triángulo rectángulo, pero aquí se da como 2, lo cual no puede ser correcto dada la longitud de los otros lados del triángulo (4 y \( \sqrt{20} \)). Asumiendo que este fue un error en la imagen, vamos a proceder como si el 2 fuera el lado opuesto al ángulo α en lugar de la hipotenusa, lo cual tendría más sentido. La tangente de un ángulo en un triángulo rectángulo se define como el lado opuesto dividido por el lado adyacente, entonces: \[ tan(\alpha) = \frac{lado\ opuesto}{lado\ adyacente} = \frac{2}{\sqrt{20}} = \frac{2}{2\sqrt{5}} = \frac{1}{\sqrt{5}} \] Pero necesitamos encontrar \( tan\left( \frac{\pi}{2} - \alpha \right) \). La función tangente es co-función de la cotangente. Esto significa que: \[ tan\left( \frac{\pi}{2} - \alpha \right) = cot(\alpha) \] La cotangente es el recíproco de la tangente, así que: \[ cot(\alpha) = \frac{1}{tan(\alpha)} = \frac{1}{\frac{1}{\sqrt{5}}} = \sqrt{5} \] Entonces, \( tan\left( \frac{\pi}{2} - \alpha \right) = \sqrt{5} \).
Finding Length of Segment in Triangle using Proportionality Theorem
The image shows a large triangle \( \triangle RQU \) with a line segment \( \overline{ST} \) cutting it proportionally, which means \( \overline{ST} \) is parallel to \( \overline{QU} \). When a line segment cuts two sides of a triangle proportionally, the Triangle Proportionality Theorem (also known as the Basic Proportionality Theorem or Thales' Theorem) states that it divides those sides proportionally. Given in the diagram: - The length of \( \overline{RQ} \) is 16. - The length of \( \overline{QS} \) is 5. - The length of \( \overline{TU} \) is 8. We need to find the length of \( \overline{RS} \). Let's denote the length of \( \overline{RS} \) as \( x \). So \( \overline{RT} = x + 5 \). According to the Triangle Proportionality Theorem: \[ \frac{RQ}{QS} = \frac{RT}{TU} \] Now we can plug the values into the equation: \[ \frac{16}{5} = \frac{x + 5}{8} \] Now, solve for \( x \): \[ 128 = 5x + 25 \] Subtract 25 from both sides: \[ 128 - 25 = 5x \] \[ 103 = 5x \] Now divide both sides by 5 to solve for \( x \): \[ x = \frac{103}{5} \] \[ x = 20.6 \] So, the length of \( \overline{RS} \) is 20.6.
Calculating Perimeter of a Right Triangle
The image shows a right triangle with the legs measuring 48 yards and 60 yards. To find the perimeter of the triangle, we need to find the lengths of all three sides and add them together. We already have the lengths of the two legs: Leg 1: 48 yards Leg 2: 60 yards To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right 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): c^2 = a^2 + b^2 Here, c is the length of the hypotenuse, and a and b are the lengths of the other two sides. Plugging in the provided values, we get: c^2 = 48^2 + 60^2 c^2 = 2304 + 3600 c^2 = 5904 Taking the square root of both sides to solve for c: c = sqrt(5904) c = 76.8 yards Now we can find the perimeter (P) by adding the lengths of all three sides: P = 48 yards + 60 yards + 76.8 yards P = 184.8 yards So, the perimeter of the triangle is 184.8 yards. If necessary, we round to the nearest tenth, but since 184.8 is already to the nearest tenth, this is our final answer.
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