Final answer:
To find the two acute angles in a right triangle where one is eight times the other, we set up the equation a + 8a = 90, solve for a to find the smaller angle as 10 degrees, then multiply by eight to find the larger angle as 80 degrees.
Step-by-step explanation:
The question involves finding the measures of two acute angles in a right triangle, where one acute angle is eight times the measure of the other. In a right triangle, the sum of the three angles is 180 degrees. Since one angle is a right angle (90 degrees), the sum of the two acute angles must be 90 degrees as well. Let's denote the smaller acute angle as a degrees. Therefore, the larger acute angle would be 8a degrees. We can set up the equation a + 8a = 90 to find the measures of the two angles.
Combining like terms, we get 9a = 90. Dividing both sides by 9, we find that a = 10 degrees. Consequently, the larger acute angle, being eight times larger, is 8a = 8 × 10 = 80 degrees. Thus, the measures of the acute angles in the right triangle are 10 degrees and 80 degrees.