Example Question - solving triangles
Here are examples of questions we've helped users solve.
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.
Solving Triangles with Given Ratios
The image shows two triangles, with a given statement that "The ratio of the LARGER triangle to the SMALLER triangle is \( \frac{3}{5} \)." To use this information to solve the problem, we must understand that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. In this case, the larger triangle has sides of lengths 21 and 18, but we are not given the corresponding sides of the smaller triangle. However, we do know one of the sides of the smaller triangle, which is 5. Since we know the ratio of the areas is \( \frac{3}{5} \), we can set up an equation using the side of the smaller triangle that we know: \[ \left( \frac{\text{Side of larger triangle}}{\text{Side of smaller triangle}} \right)^2 = \frac{3}{5} \] We can use either 21 (corresponding to the larger side) or 18 (corresponding to the smaller side) for the "Side of larger triangle" in the equation. Let's use 18 as it corresponds with the side length of 5 in the smaller triangle: \[ \left( \frac{18}{5} \right)^2 = \frac{3}{5} \] To find the "Side of larger triangle" that corresponds to the smaller triangle's side of length 5, we can use the square root to get: \[ \left( \frac{18}{5} \right) = \sqrt{\frac{3}{5}} \] \[ \frac{18}{5} = \frac{\sqrt{3}}{\sqrt{5}} \] Now, to solve for the corresponding side in the larger triangle, we cross-multiply: \[ 5\sqrt{3} = 18\sqrt{5} \] \[ \text{Side of larger triangle} = \frac{18\sqrt{5}}{\sqrt{3}} \] \[ \text{Side of larger triangle} = \frac{18\sqrt{3}\sqrt{5}}{3} \] \[ \text{Side of larger triangle} = 6\sqrt{15} \] So the side of the larger triangle that corresponds to the side of length 5 in the smaller triangle has a length of \( 6\sqrt{15} \), which would be the correct value to fill in the blank space in the image.
Solving Triangles with Parallel Lines
Claro, vamos a resolver el problema que se muestra en la imagen usando conceptos de geometría. En este problema, nos muestran un triángulo \( ABC \) en el cual el segmento \( \overline{DE} \) es paralelo al segmento \( \overline{BC} \), y se indican algunas medidas de longitudes: \( AE = 5 \) cm y \( EC = 4 \) cm. Nos piden encontrar el valor de \( x \) en función de \( e \). Usaremos la propiedad de los triángulos semejantes que dice que si una línea es paralela a uno de los lados de un triángulo y corta a los otros dos lados, entonces divide a esos dos lados en segmentos proporcionales. El triángulo pequeño \( ADE \) es semejante al triángulo grande \( ABC \) porque tienen dos ángulos congruentes (causados por la línea paralela y los ángulos correspondientes), por lo que podemos plantear la relación: \[ \frac{AE}{EC} = \frac{AD}{DB} \] Dado que \( AE = 5 \) cm y \( EC = 4 \) cm, la relación queda: \[ \frac{5}{4} = \frac{AD}{DB} \] Ahora, podemos expresar \( DB \) como la longitud total de \( BC \) menos la longitud de \( AD \), es decir, \( DB = (e+5) - AD \), donde usamos que \( AD = 5 + x \) y \( BC = e \). Sin embargo, notemos que para encontrar \( x \) en función de \( e \), realmente no necesitamos calcular \( DB \) ya que \( BC \) y \( AD \) están en términos de \( x \) y \( e \). Ahora calculamos \( x \) en función de \( e \) utilizando la semejanza de triángulos: \[ \frac{5}{4} = \frac{5 + x}{e} \] Multiplicaremos en cruz para resolver por \( x \): \[ 5e = 4(5 + x) \] \[ 5e = 20 + 4x \] Restamos 20 de ambos lados de la ecuación para despejar \( 4x \): \[ 4x = 5e - 20 \] Finalmente, dividimos ambos lados entre 4 para obtener \( x \): \[ x = \frac{5e - 20}{4} \] Esa es la expresión de \( x \) en función de \( e \). En las opciones proporcionadas, este resultado corresponde a: \[ D) \frac{5e - 20}{4} \] Por lo tanto, la respuesta correcta es la opción D.
Solving Triangles Using Angle Sum Property
En la imagen se presenta un triángulo con sus tres ángulos marcados. Para resolver la pregunta, es necesario recordar que la suma de los ángulos internos de un triángulo siempre es igual a 180 grados. El problema nos da tres expresiones para los ángulos del triángulo: x + 1, x y x + 2. Podemos establecer la siguiente ecuación sumando estas tres expresiones y estableciendo su suma igual a 180 grados: x + 1 + x + x + 2 = 180 Al simplificar la ecuación, combinamos los términos semejantes: 3x + 3 = 180 Ahora restamos 3 de ambos lados de la ecuación: 3x = 177 Y luego dividimos ambos lados entre 3 para despejar x: x = 177 / 3 x = 59 Con este valor, podemos encontrar la medida de cada ángulo del triángulo: Ángulo rojo (x + 1): 59 + 1 = 60 grados Ángulo verde claro (x): 59 grados Ángulo verde oscuro (x + 2): 59 + 2 = 61 grados Y con esto hemos resuelto el valor de x y también la medida de los ángulos del triángulo en la imagen.
Solving Right-Angled Triangles Using the Pythagorean Theorem
The image shows two right-angled triangles that are not drawn to scale. For both triangles, we can use the Pythagorean theorem to solve for the missing side lengths. The Pythagorean theorem states that for a right-angled triangle with sides a, b, and hypotenuse c, the following equation holds true: a^2 + b^2 = c^2. For the first triangle: We need to find the length of the missing side that we'll call a. We have the lengths of the other side (b) and the hypotenuse (c). The hypotenuse is the longest side, which is 5 units, and the other side given is 4 units. Using the Pythagorean theorem, we get: a^2 + 4^2 = 5^2 a^2 + 16 = 25 a^2 = 25 - 16 a^2 = 9 a = √9 a = 3 The length of the missing side is 3 units, corresponding to option B. For the second triangle: The missing side that we need to find is the hypotenuse, which we'll call c. The two sides given are a = 24 units and b = 7 units. Plugging these values into the Pythagorean theorem gives us: 24^2 + 7^2 = c^2 576 + 49 = c^2 625 = c^2 c = √625 c = 25 The length of the missing side, which is the hypotenuse, is 25 units, and this corresponds to option B. So, the answers are: Question 4: B. 3 Question 5: B. 25
Solving Unknown Angles in a Triangle
The image shows a diagram of a triangle with one of its angles labeled as 26 degrees. It seems the problem is asking you to find the unknown angles x, y, and z of the triangle. To solve for the unknown angles, we would use the fact that the sum of the angles in any triangle is 180 degrees. Given that one angle is 26 degrees, we can find the sum of the other two angles: 180 degrees (sum of all angles in a triangle) - 26 degrees (given angle) = 154 degrees Now we have an equation for the combined measure of angles x and y: x + y = 154 degrees However, we can't find the exact measures of x and y without additional information. For angle z, it's not clear from the diagram whether z is supposed to be an angle on the inside of the triangle or an external angle, nor is there an indication of the relationship between z and other angles. If z is an external angle adjacent to the 26-degree angle, then: z = 180 degrees - 26 degrees (since the sum of a straight line is 180 degrees) z = 154 degrees But if z is an angle within the triangle, we would need to know its relationship to the other angles to solve for it. Please check if there is more information provided or if there is a specific question you need to answer.
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