Acute Triangle and Equation Problems

This lesson covers solving special equations, focusing on linear Diophantine equations, a birth date problem, and an acute angle triangle problem. The instructor explains how to find integer solutions for equations where coefficients are co-prime, and demonstrates the process through examples. The lesson also includes a problem involving calculating angles in a triangle, ensuring all angles are less than 90 degrees.

What is the first step in solving equations where the coefficients are co-prime?

• Find the greatest common divisor

• Find one solution

• Multiply both sides by a constant

• Divide both sides by a constant

In the equation 7x + 13y = 186, what is the first solution found for y?

• y = 1

• y = 2

• y = 3

• y = 4

How does the value of x change when finding additional solutions for 7x + 13y = 186?

• Increases by 7

• Decreases by 13

• Increases by 13

• Decreases by 7

What is the total number of positive integer solutions for the equation 7x + 13y = 186?

• Three

• Two

• One

• Four

In the birth date problem, what is the equation used to find the date and month?

• 12d + 31m = 425

• 31d - 12m = 425

• 12d - 31m = 425

• 31d + 12m = 425

What is the month of birth if the equation 12d + 31m = 425 is satisfied?

• October

• November

• December

• January

In an acute triangle, what is the sum of all angles?

• 90 degrees

• 180 degrees

• 270 degrees

• 360 degrees

If one angle in an acute triangle is 13 times another, what must be true for the angles?

• All angles are less than 90 degrees

• All angles are equal

• One angle is greater than 90 degrees

• One angle is exactly 90 degrees

What is the largest angle possible in the given acute triangle problem?

• 75 degrees

• 85 degrees

• 60 degrees

• 90 degrees

What is the minimum value of y for the angle relationship 13x = 17y in an acute triangle?

• 13

• 39

• 26

• 52