The measure of one acute angle of a right triangle is twelve more than three times the measure than the other acute angle. Find the measure of each acute angle of the right triangle.
a + 3a + 12 = 90
Solve for a to find the smaller angle.
a + (3a+12) = 90
4 a = 78
a = 19.5 degrees
90 - 19.5 = 70.5
check
3(19.5) + 12 = 70.5 ok
To solve this problem, let's first assign variables to the unknown angles of the right triangle.
Let's say that one acute angle of the right triangle is represented by x degrees, and the other acute angle is represented by y degrees.
According to the given information, one acute angle is twelve more than three times the other acute angle. We can express this using an equation:
x = 3y + 12
Now, we know that the sum of all angles in a triangle is 180 degrees. In a right triangle, one angle is 90 degrees, so the sum of the two acute angles is 180 - 90 = 90 degrees.
Therefore, we can set up another equation:
x + y = 90
Now, we have a system of two equations:
x = 3y + 12
x + y = 90
We can solve this system of equations to find the values of x and y.
1. Substitute the expression for x from the first equation into the second equation:
(3y + 12) + y = 90
2. Simplify the equation by combining like terms:
4y + 12 = 90
3. Subtract 12 from both sides of the equation:
4y = 78
4. Divide both sides of the equation by 4:
y = 19.5
5. Substitute the value of y back into the first equation to find x:
x = 3(19.5) + 12
x = 58.5 + 12
x = 70.5
Therefore, the measure of one acute angle of the right triangle is 70.5 degrees, and the measure of the other acute angle is 19.5 degrees.