The angles in a triangle are in the ratio $2: 1: 6$. Calculate the size of each angle.
Answer (2)
Represent the angles as 2 x , x , and 6 x based on the given ratio.
Formulate the equation 2 x + x + 6 x = 180 using the fact that the sum of angles in a triangle is 180 degrees.
Solve for x to find x = 20 .
Calculate each angle: 2 x = 40 degrees, x = 20 degrees, and 6 x = 120 degrees, so the angles are 40 , 20 , 120 .
Explanation
Analyze the problem Let's analyze the problem. We are given that the angles of a triangle are in the ratio 2 : 1 : 6 . We need to find the measure of each angle. We know that the sum of the angles in any triangle is 180 degrees.
Set up the equation Let the angles be 2 x , x , and 6 x . Since the sum of the angles in a triangle is 180 degrees, we can write the equation: 2 x + x + 6 x = 180
Simplify the equation Combining the terms on the left side, we get: 9 x = 180
Solve for x Now, we solve for x by dividing both sides of the equation by 9 :
x = 9 180 = 20
Calculate the angles Now that we have the value of x , we can find the measure of each angle:
First angle: 2 x = 2 ( 20 ) = 40 degrees Second angle: x = 20 degrees Third angle: 6 x = 6 ( 20 ) = 120 degrees
State the final answer Therefore, the angles of the triangle are 40 degrees, 20 degrees, and 120 degrees. Let's check if the sum of the angles is 180 degrees: 40 + 20 + 120 = 180
Examples
Understanding angles and their ratios is crucial in many real-world applications. For example, architects use these concepts when designing buildings to ensure structural stability and aesthetic appeal. Similarly, in navigation, pilots and sailors use angles to determine their course and direction. In art, understanding angles and proportions helps artists create realistic and visually appealing drawings and paintings. Even in sports, athletes use angles to optimize their performance, such as in golf or basketball.
The angles in the triangle are 40 degrees, 20 degrees, and 120 degrees, calculated using the ratio 2:1:6. We represented the angles as 2 x , x , and 6 x , combined them to equal 180 degrees, and solved for x to find each angle. The solutions confirm that the sum of the angles is indeed 180 degrees.
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