Question

1. given isosceles triangle abc as shown below where \\( \overline{ab} \cong \overline{ac} \\) and \\( ab = 7x + 8 \\) and \\( ac = 4x + 26 \\), find the length of bc if \\( bc = 5x - 2 \\).

equation:

\\( x = \underline{quadquad} \\) \\( bc = \underline{quadquad}

1. given \\( \triangle abc \\), write an equation and solve for \\( x \\) to find \\( m \angle a \\).

equation:

\\( x = \underline{quadquad} \\) \\( m \angle a = \underline{quadquad} \\)

Explanation:

Problem 17

Step 1: Set AB equal to AC (isosceles triangle property)

Since \( AB \cong AC \), their lengths are equal. So, \( 7x + 8 = 4x + 26 \)

Step 2: Solve for x

Subtract \( 4x \) from both sides: \( 7x - 4x + 8 = 26 \)
Simplify: \( 3x + 8 = 26 \)
Subtract 8 from both sides: \( 3x = 26 - 8 \)
\( 3x = 18 \)
Divide by 3: \( x = \frac{18}{3} = 6 \)

Step 3: Find length of BC

Substitute \( x = 6 \) into \( BC = 5x - 2 \)
\( BC = 5(6) - 2 = 30 - 2 = 28 \)

Step 1: Use triangle angle sum property (sum of angles in a triangle is \( 180^\circ \))

The angles are \( \angle A = 4x + 16^\circ \), \( \angle C = 2x + 2^\circ \), \( \angle B = 2x - 14^\circ \). So,
\( (4x + 16) + (2x + 2) + (2x - 14) = 180 \)

Step 2: Simplify and solve for x

Combine like terms: \( 4x + 2x + 2x + 16 + 2 - 14 = 180 \)
\( 8x + 4 = 180 \)
Subtract 4: \( 8x = 180 - 4 = 176 \)
Divide by 8: \( x = \frac{176}{8} = 22 \)

Step 3: Find \( m\angle A \)

Substitute \( x = 22 \) into \( 4x + 16 \):
\( 4(22) + 16 = 88 + 16 = 104^\circ \)

Answer:

Equation: \( 7x + 8 = 4x + 26 \)
\( x = 6 \)
\( BC = 28 \)

Problem 18