Example Question - triangle sides length
Here are examples of questions we've helped users solve.
Finding the Length of Small Equilateral Triangle Inside a Larger Equilateral Triangle
Para resolver el problema en la imagen, necesitamos encontrar el valor de \( x \). La figura muestra un triángulo equilátero (todos los lados son iguales) con los lados marcados como 8, y dentro de este triángulo hay otro triángulo equilátero más pequeño cuyo lado se denota con \( x \). Para resolver \( x \), podemos utilizar el teorema de Pitágoras en uno de los triángulos rectángulos que se forman al conectar los puntos medios de los lados del triángulo grande. Si trazamos una línea desde un vértice del triángulo grande al punto medio del lado opuesto, esta línea sería una altura del triángulo grande y también sería la mitad de un lado del triángulo pequeño. La altura \( h \) del triángulo grande se puede obtener aplicando el teorema de Pitágoras en uno de estos triángulos rectángulos, donde la hipotenusa es un lado del triángulo grande (8 unidades), y uno de los catetos es la mitad de un lado del triángulo pequeño (\( x/2 \)). El otro cateto es la altura \( h \) misma. La ecuación sería: \( h^2 + (x/2)^2 = 8^2 \) Ya que es un triángulo equilátero grande, la altura \( h \) también divide al lado opuesto en dos segmentos que suman 8 unidades. Dado que el triángulo más pequeño tiene la misma altura que uno de estos segmentos: \( h = 8 - x/2 \) Ahora podemos expresar \( h^2 \) en función de \( x \): \( (8 - x/2)^2 + (x/2)^2 = 8^2 \) Expandiendo el cuadrado de la binomio obtenemos: \( 64 - 2 \cdot 8 \cdot x/2 + (x/2)^2 + (x/2)^2 = 64 \) Simplificando la ecuación: \( 64 - 8x + x^2/4 + x^2/4 = 64 \) Combinando términos semejantes y eliminando el término constante 64 de ambos lados: \( x^2/2 - 8x = 0 \) Factorizando \( x \) obtenemos: \( x(x/2 - 8) = 0 \) Así que \( x = 0 \) o \( x/2 - 8 = 0 \). Despejando para \( x \) de la segunda ecuación, obtenemos: \( x/2 = 8 \) \( x = 16 \) Dado que el valor de \( x = 0 \) no tiene sentido en este contexto, la longitud del lado del triángulo más pequeño es \( x = 16 \) unidades.
Calculating the Length of the Hypotenuse in a Right Triangle
In this image, we see a right triangle with one side measuring 36 kilometers and another side (which is the opposite of the 90-degree angle) measuring 77 kilometers. The question asks for the length of the hypotenuse, denoted as "c." To find the length of the hypotenuse, 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. The formula is: c^2 = a^2 + b^2 Here, a = 36 km and b = 77 km, so we have: c^2 = 36^2 + 77^2 c^2 = 1296 + 5929 c^2 = 7225 Taking the square root of both sides gives us the length of the hypotenuse c: c = √7225 c ≈ 85 Now, rounding to the nearest tenth as requested, we get: c ≈ 85.0 kilometers Therefore, the length of the hypotenuse is approximately 85.0 kilometers.
Calculating the Length of the Hypotenuse of a Right-Angled Triangle
The image displays a right-angled triangle with the two legs measuring 60 meters and 80 meters, and the length of the hypotenuse labeled as "c." To find the length of the hypotenuse, we use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The Pythagorean theorem is expressed 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 the side lengths of 60 meters and 80 meters, we can plug them into the equation: \[ c^2 = 60^2 + 80^2 \] \[ c^2 = 3600 + 6400 \] \[ c^2 = 10000 \] To find \( c \), we take the square root of both sides of the equation: \[ \sqrt{c^2} = \sqrt{10000} \] \[ c = 100 \] Thus, the length of the hypotenuse is 100 meters. There is no need to round to the nearest tenth, since we have an exact value.
Calculation of Hypotenuse in a Right-Angled Triangle
The image portrays a right-angled triangle with one leg measuring 60 meters and the other leg measuring 80 meters. To find the length of the hypotenuse (`c`), we would use the Pythagorean theorem. The theorem 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 (the legs). The formula is: \[ c^2 = a^2 + b^2 \] where `a` and `b` are the lengths of the legs, and `c` is the length of the hypotenuse. Here, `a = 60 m` and `b = 80 m`, so we plug these values into the formula to get: \[ c^2 = (60)^2 + (80)^2 \] \[ c^2 = 3600 + 6400 \] \[ c^2 = 10000 \] To find `c`, we take the square root of both sides: \[ c = \sqrt{10000} \] \[ c = 100 \] Therefore, the length of the hypotenuse, `c`, is 100 meters. There is no need to round since the result is a whole number.
Calculating the Length of the Hypotenuse of a Right Triangle
The triangle shown in the image is a right triangle, and we are looking for the length of the hypotenuse, denoted as "c". According to the Pythagorean theorem, in a right 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. This can be written as: c² = a² + b² where c = length of the hypotenuse, a and b = lengths of the other two sides of the right triangle. From the image, we have: a = 60 meters, b = 80 meters. Now we plug these values into the Pythagorean theorem: c² = 60² + 80² c² = 3600 + 6400 c² = 10000 To find c, we take the square root of both sides: c = √10000 c = 100 Therefore, the length of the hypotenuse is 100 meters. Since the problem states that we should round to the nearest tenth if necessary, we can express our final answer as 100.0 meters, although it is not required in this case since 100 is a whole number.
Calculation of Hypotenuse in a Right Triangle
题目描述了一个直角三角形ABC，其中∠BAC是直角，给出了AB的长度为3，AC的长度为2。由直角三角形的性质知，BC是三角形ABC的斜边。题目要求求出BC的长。 根据勾股定理（直角三角形的两条直角边的平方和等于斜边的平方），可以设BC的长为c，得到： AB² + AC² = BC² 3² + 2² = c² 9 + 4 = c² 13 = c² c = √13 所以BC的长度为√13。
Determining Right Triangle
To determine if a triangle is a right triangle when you know the lengths of its sides, you can use the Pythagorean theorem, which states that in a right 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 is expressed as: a² + b² = c² Where c is the hypotenuse and a and b are the other two sides. In the question, the sides are given as 22, 32, and 55. Assuming the hypotenuse is 55 (because it is the longest side which is always opposite the right angle in a right triangle), then: a² + b² should equal c². Let's check: 22² + 32² = 55² 484 + 1024 = 3025 1508 ≠ 3025 The sum of the squares of the two shorter sides does not equal the square of the longest side, so this triangle is not a right triangle. The answer to the problem is: No, it is not a right triangle.
Calculating Perimeter of a Triangle
To find the perimeter of a triangle, you simply add up the lengths of all three sides. Since the triangle in question has each side measuring 0.75 inches, you'd calculate the perimeter as follows: Perimeter = side_1 + side_2 + side_3 Perimeter = 0.75 inches + 0.75 inches + 0.75 inches Perimeter = 3 * 0.75 inches Perimeter = 2.25 inches So, the perimeter of the triangle is 2.25 inches. The correct answer to the question in the image is 2.25 inches.
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