Example Question - triangle area formula
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Calculating Perimeter and Area of a Triangle
Para resolver este problema, primero hay que calcular el perímetro del triángulo. El perímetro es la suma de las longitudes de todos los lados del triángulo. Según la imagen, las longitudes de los lados son: - \( AB = \frac{3}{4} \) pulgadas - \( BC = \frac{3}{8} \) pulgadas - \( AC = \frac{5}{8} \) pulgadas Sumamos las longitudes para obtener el perímetro: \( P = AB + BC + AC \) \( P = \frac{3}{4} + \frac{3}{8} + \frac{5}{8} \) Para sumar las fracciones, necesitamos un denominador común, que en este caso es 8. Convertimos \( \frac{3}{4} \) a octavos multiplicando el numerador y el denominador por 2: \( \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \) Ahora podemos sumar todas las fracciones: \( P = \frac{6}{8} + \frac{3}{8} + \frac{5}{8} \) \( P = \frac{6 + 3 + 5}{8} \) \( P = \frac{14}{8} \) Simplificamos la fracción dividiendo tanto el numerador como el denominador entre 2: \( P = \frac{14 ÷ 2}{8 ÷ 2} \) \( P = \frac{7}{4} \) pulgadas Así que el perímetro del triángulo es \( \frac{7}{4} \) pulgadas o 1 \( \frac{3}{4} \) pulgadas. Para encontrar el área, necesitamos la base y la altura del triángulo. En la imagen, \( AC \) es la base y \( AD \) es la altura. La fórmula para la área de un triángulo es: \( A = \frac{base \times altura}{2} \) La base \( AC = \frac{5}{8} \) pulgadas y la altura \( AD = 1 \) pulgada, entonces: \( A = \frac{\frac{5}{8} \times 1}{2} \) \( A = \frac{5}{8} \times \frac{1}{2} \) \( A = \frac{5}{16} \) pulgadas cuadradas Por lo tanto, el área del triángulo es \( \frac{5}{16} \) pulgadas cuadradas.
Calculating Perimeter and Area of a Triangle
Para resolver el problema, primero calcularemos el perímetro del triángulo ABC y luego calcularemos el área. El perímetro de un triángulo se calcula sumando la longitud de sus tres lados. Tenemos las longitudes de los lados AB, BC y AC: AB = \( \frac{5}{8} \) pulg + \( \frac{1}{2} \) pulg = \( \frac{5}{8} \) pulg + \( \frac{4}{8} \) pulg = \( \frac{9}{8} \) pulg BC = \( \frac{1}{4} \) pulg + \( \frac{3}{8} \) pulg = \( \frac{2}{8} \) pulg + \( \frac{3}{8} \) pulg = \( \frac{5}{8} \) pulg AC ya viene dado como 1 pulg. Así que el perímetro P será la suma de estas longitudes: P = AB + BC + AC = \( \frac{9}{8} \) pulg + \( \frac{5}{8} \) pulg + 1 pulg Para sumar las fracciones, convertimos 1 pulg en octavos, lo cual es \( \frac{8}{8} \) pulg: P = \( \frac{9}{8} \) pulg + \( \frac{5}{8} \) pulg + \( \frac{8}{8} \) pulg = \( \frac{22}{8} \) pulg = 2 \( \frac{6}{8} \) pulg = 2 \( \frac{3}{4} \) pulg (simplificando la fracción). Ahora calculamos el área A del triángulo, que es: A = \( \frac{1}{2} \) base * altura La base del triángulo es AC y la altura es BD. Base (AC) = 1 pulg Altura (BD) = \( \frac{5}{8} \) pulg Por lo tanto, el área A es: A = \( \frac{1}{2} \) * 1 pulg * \( \frac{5}{8} \) pulg = \( \frac{5}{16} \) pulg² Aplicando estas operaciones, hemos encontrado que el perímetro del triángulo es 2 \( \frac{3}{4} \) pulg y el área es \( \frac{5}{16} \) pulg².
Calculating Triangle Perimeter and Area
Para resolver la pregunta, primero vamos a calcular el perímetro del triángulo y luego el área. El perímetro de un triángulo es simplemente la suma de la longitud de sus tres lados. Entonces, sumando las longitudes de los lados AB, BC y CA, tendremos: \[ AB + BC + CA = \frac{5}{8} + \frac{3}{2} + \frac{1}{4} \] Para sumar estas fracciones, necesitamos que todas tengan el mismo denominador. Tomamos el mínimo común denominador, que es 8, y obtenemos: \[ AB + BC + CA = \frac{5}{8} + \frac{12}{8} + \frac{2}{8} = \frac{5 + 12 + 2}{8} = \frac{19}{8} \] Por lo tanto, el perímetro del triángulo es \( \frac{19}{8} \) pulgadas. Para calcular el área, utilizamos la fórmula para el área de un triángulo: \[ Área = \frac{base \times altura}{2} \] La base del triángulo es CD y la altura es BB'. La longitud de CD es \( \frac{3}{8} \) pulgadas y la altura BB' (que es la misma que AB) es \( \frac{5}{8} \) pulgadas. Sustituyendo en la fórmula, obtenemos: \[ Área = \frac{\left (\frac{3}{8} \right ) \times \left (\frac{5}{8} \right )}{2} \] Multiplicamos las fracciones del numerador: \[ Área = \frac{3 \times 5}{8 \times 8 \times 2} \] \[ Área = \frac{15}{128} \] Por lo tanto, el área del triángulo es \( \frac{15}{128} \) pulgadas cuadradas. En resumen, el perímetro del triángulo es \( \frac{19}{8} \) pulgadas y el área es \( \frac{15}{128} \) pulgadas cuadradas.
Calculating the Area of a Triangle
The image presents a formula for calculating the area of a triangle. The formula given is: \[ A = \frac{1}{2} b h \] Where: A represents the area of the triangle, b represents the base of the triangle, and h represents the height of the triangle. To solve for the area of a specific triangle, you would need to know the measurements of the base and the height of the triangle. Once you have those measurements, you would plug them into the formula, multiply the base (b) by the height (h), and then multiply that product by \( \frac{1}{2} \) (or divide by 2) to find the area of the triangle.
Calculating Area of a Right-Triangle
The image shows a right-angled triangle with a base length of 23 cm and a height of 13 cm. To calculate the area of this triangle, you can use the formula for the area of a triangle: Area = (base * height) / 2 Plugging in the given values: Area = (23 cm * 13 cm) / 2 Area = (299 cm²) / 2 Area = 149.5 cm² So, the area of the triangle is 149.5 square centimeters.
Calculating Area of a Triangle using Shoelace Formula
To find the area of the triangle with vertices \((-1, 4)\), \((3, 2)\), and \((5, 3)\), you can use the Shoelace formula (also called the Gauss area formula). This formula can be applied when you have the coordinates of the triangle's vertices. The Shoelace formula states that the area of the triangle \( A \) can be found by: \[ A = \frac{1}{2} \left| (x_1y_2 + x_2y_3 + x_3y_1) - (y_1x_2 + y_2x_3 + y_3x_1) \right| \] Where \( (x_1, y_1), (x_2, y_2), \) and \( (x_3, y_3) \) are the coordinates of the vertices. Let's plug in the coordinates: \[ x_1 = -1, y_1 = 4 \\ x_2 = 3, y_2 = 2 \\ x_3 = 5, y_3 = 3 \] \[ A = \frac{1}{2} | ((-1) \cdot 2 + 3 \cdot 3 + 5 \cdot 4) - (4 \cdot 3 + 2 \cdot 5 + 3 \cdot (-1)) | \] Now multiply and add the corresponding terms: \[ A = \frac{1}{2} | ((-2) + 9 + 20) - (12 + 10 - 3) | \] Simplify the terms inside the absolute value: \[ A = \frac{1}{2} | 27 - 19 | \] \[ A = \frac{1}{2} | 8 | \] \[ A = \frac{1}{2} \cdot 8 \] \[ A = 4 \] So the area of the triangle is 4 square units.
Calculating Height of a Triangle
The area of a triangle is given by the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Given: - The area of the triangle \( A = 48 \, \text{cm}^2 \) - The base of the triangle \( b = 12 \, \text{cm} \) Now plug in the values and solve for the height \( h \): \[ 48 = \frac{1}{2} \times 12 \times h \] To find the height, perform the following steps: \[ 48 = 6 \times h \] \[ h = \frac{48}{6} \] \[ h = 8 \] The height of the triangle is \( 8 \, \text{cm} \).
Calculating Triangle Area
The image shows a triangle with a base length of 8.4 inches and a height indicated by a dashed line of 9 inches. To find the area of a triangle, you can use the formula: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] Plugging in the given values: \[ \text{Area} = \frac{1}{2} \times 8.4 \, \text{inches} \times 9 \, \text{inches} \] \[ \text{Area} = 4.2 \times 9 \] \[ \text{Area} = 37.8 \, \text{square inches} \] Thus, the area of the triangle is 37.8 square inches.
Finding Included Angle of a Triangle
The question in the image is regarding finding the measure of the included angle of a triangle, given the area and the lengths of two sides. The formula to find the area of a triangle when two sides and the included angle are known is: \[ \text{Area} = \frac{1}{2} ab \sin(C) \] Where: - \(a\) and \(b\) are the lengths of the two sides, - \(C\) is the included angle, - \( \text{Area} \) is the area of the triangle. Given: - \(a = 55 \), - \(b = 52 \), - \( \text{Area} = 986 \). Let's find the angle \(C\): \[ 986 = \frac{1}{2} \times 55 \times 52 \times \sin(C) \] \[ 986 = \frac{1}{2} \times 2860 \times \sin(C) \] \[ \sin(C) = \frac{986}{1430} \] \[ \sin(C) = 0.68951... \] Now, take the inverse sine (arc sine) of both sides to find the angle \(C\): \[ C = \arcsin(0.68951...) \] Using a calculator, you'll find that: \[ C \approx 43.60282^{\circ} \] Therefore, the measure of the included angle, rounded to the nearest tenth of a degree, is approximately \(43.6^{\circ}\).
Calculating Area of Triangle with Given Base and Height
The question shows a triangular sail with a base labeled as "1z" and a height labeled as "2z." The problem states that the base of the white sail is 10 feet. The formula for the area of a triangle is given as Area = 1/2 (Base x Height). From the problem statement, we know that 1z = 10 feet. Since the height of the sail is labeled as "2z," this means the height is 2 times 10 feet, which is 20 feet. Now, we can use the formula to calculate the area with the given dimensions: Area = 1/2 (Base x Height) Area = 1/2 (10 feet x 20 feet) Area = 1/2 (200 square feet) Area = 100 square feet So the area of the sail is 100 square feet, which corresponds to option B.
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