Question

hw 7: relationships in triangles
list the angles of \\( \triangle a b c \\) in order from least to greatest if \\( a b = 6 x - 35, b c = 4 x + 11, a c = x + 29 \\), and the perimeter of \\( \triangle a b c = 192 \\).

Explanation:

Step1: Find the value of \( x \) using the perimeter formula.

The perimeter of a triangle is the sum of its three sides. So, \( AB + BC + AC = 192 \). Substituting the given expressions:
\( (6x - 35) + (4x + 11) + (x + 29) = 192 \)
Combine like terms:
\( 6x + 4x + x - 35 + 11 + 29 = 192 \)
\( 11x + 5 = 192 \)
Subtract 5 from both sides:
\( 11x = 192 - 5 = 187 \)
Divide by 11:
\( x = \frac{187}{11} = 17 \)

Step2: Calculate the lengths of each side.

• \( AB = 6x - 35 = 6(17) - 35 = 102 - 35 = 67 \)
• \( BC = 4x + 11 = 4(17) + 11 = 68 + 11 = 79 \)
• \( AC = x + 29 = 17 + 29 = 46 \)

Step3: Recall the triangle angle - side relationship.

In a triangle, the larger the side opposite an angle, the larger the angle. So we need to determine which angle is opposite each side:

• Side \( AC \) is opposite \( \angle B \)
• Side \( AB \) is opposite \( \angle C \)
• Side \( BC \) is opposite \( \angle A \)

Now, order the sides from least to greatest: \( AC = 46 \), \( AB = 67 \), \( BC = 79 \)

Step4: Order the angles based on the side - angle relationship.

Since \( AC < AB < BC \), the angles opposite them will follow the same order. The angle opposite \( AC \) is \( \angle B \), the angle opposite \( AB \) is \( \angle C \), and the angle opposite \( BC \) is \( \angle A \). So the order of angles from least to greatest is \( \angle B \), \( \angle C \), \( \angle A \)

Answer:

\( \angle B \), \( \angle C \), \( \angle A \)