Question

in triangle abc, ( mangle abc=(4x - 12)^{circ} ) and ( mangle acb=(2x + 26)^{circ} ). yin says that if ( x = 19 ), the triangle must be equilateral. is he correct? justify your answer.

Explanation:

Step1: Calculate \( m\angle ABC \) when \( x = 19 \)

Substitute \( x = 19 \) into \( m\angle ABC=(4x - 12)^\circ \).
\( m\angle ABC=4\times19 - 12=76 - 12 = 64^\circ \)

Step2: Calculate \( m\angle ACB \) when \( x = 19 \)

Substitute \( x = 19 \) into \( m\angle ACB=(2x + 26)^\circ \).
\( m\angle ACB=2\times19+26 = 38 + 26=64^\circ \)

Step3: Calculate \( m\angle BAC \)

In a triangle, the sum of interior angles is \( 180^\circ \). Let \( m\angle BAC = y \). Then \( y+64 + 64=180 \).
\( y=180-(64 + 64)=180 - 128 = 52^\circ \)

Step4: Determine if the triangle is equilateral

An equilateral triangle has all angles equal to \( 60^\circ \). Here, the angles are \( 64^\circ \), \( 64^\circ \), and \( 52^\circ \), which are not all equal.

Answer:

Yin is not correct. When \( x = 19 \), \( m\angle ABC = 64^\circ \), \( m\angle ACB=64^\circ \), and \( m\angle BAC = 52^\circ \). Since not all angles are equal (an equilateral triangle requires all angles to be \( 60^\circ \)), the triangle is not equilateral.