The ratio of the measures of the acute angles of a right triangle is 3:2. What is the measure of the smaller acute angle?
To find the measure of the smaller acute angle in a right triangle with a ratio of 3:2, we need to use the concept of ratios and proportions.
In a right triangle, one of the acute angles is always 90 degrees (the right angle). Let's assume the other two angles as X and Y, where X is the smaller angle and Y is the larger angle.
According to the given ratio, X:Y = 3:2. This means, X is 3 times smaller than Y.
We can set up a proportion using this information:
X/Y = 3/2
Now, cross-multiply:
2X = 3Y
Divide both sides by 2 to solve for X:
X = (3Y)/2
Since we know the sum of the angles in a triangle is always 180 degrees, we can write the equation:
X + Y + 90 = 180
Now substitute the value of X from the proportion into the equation:
(3Y)/2 + Y + 90 = 180
Simplify the equation:
(3Y + 2Y + 180)/2 = 180
Combine like terms:
5Y + 180 = 360
Subtract 180 from both sides:
5Y = 180
Divide both sides by 5:
Y = 36
Now substitute the value of Y into the proportion to find X:
X = (3 * 36) / 2
X = 108 / 2
X = 54
Therefore, the measure of the smaller acute angle in the right triangle is 54 degrees.
let the angles be 3x and 2x
their sum = 90°
5x = 90
x = 18
the smaller angles is 2x = 36°