Problem 1216 The three angles of a triangle a... [FREE SOLUTION]

Chapter 42: Problem 1216

The three angles of a triangle are together equal to \(180^{\circ}\). The smallest angle is half as large as the largest one, and the sum of the largest and smallest angles is twice the third angle. Find the three angles.

Short Answer

Expert verified

The three angles of the triangle are x = 40 degrees, y = 60 degrees, and z = 80 degrees.

Step by step solution

Variables and Equations

We will use three variables to represent the triangle's angles: Let x be the smallest angle. Let y be the third angle. Let z be the largest angle. We know the following three properties about the triangle: a) The sum of all angles is 180 degrees: x + y + z = 180 b) The smallest angle is half of the largest angle: x = 0.5z c) The sum of the smallest and largest angles is twice the third angle: x + z = 2y

Solve the first equation for x

To eliminate one of the variables, we'll solve the first equation for x: x = 180 - y - z

Substitute x in equations b and c

Now we'll substitute the expression of x from equation a into equations b and c: 180 - y - z = 0.5z (equation b) 180 - y - z + z = 2y (equation c)

Solve equation b for y

From equation b, we can simplify and solve for y: 180 - y - z = 0.5z 180 - y = 1.5z y = 180 - 1.5z

Substitute y in equation c

Now we'll substitute the expression of y from equation b into equation c: 180 - (180 - 1.5z) = 2(180 - 1.5z)

Solve equation c for z

Now we can simplify and solve equation c for z: 180 - 180 + 1.5z = 360 - 3z 1.5z + 3z = 360 4.5z = 360 Divide both sides by 4.5: z = 80

Calculate x and y

Now that we have the value of z, we can find the values of x and y: x = 0.5z = 0.5 * 80 = 40 y = 180 - 1.5z = 180 - 1.5 * 80 = 60 The three angles are x = 40 degrees, y = 60 degrees, and z = 80 degrees.

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91影视

The three angles of a triangle are together equal to \(180^{\circ}\). The smallest angle is half as large as the largest one, and the sum of the largest and smallest angles is twice the third angle. Find the three angles.

Short Answer

Step by step solution

Variables and Equations

Solve the first equation for x

Substitute x in equations b and c

Solve equation b for y

Substitute y in equation c

Solve equation c for z

Calculate x and y

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