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 ∠ABC

Substitute x=19 into (4x-12)°:
$4(19) - 12 = 76 - 12 = 64^\circ$

Step2: Calculate ∠ACB

Substitute x=19 into (2x+26)°:
$2(19) + 26 = 38 + 26 = 64^\circ$

Step3: Calculate ∠BAC

Sum of angles in triangle is 180°:
$180^\circ - 64^\circ - 64^\circ = 52^\circ$

Step4: Check equilateral condition

Equilateral triangle needs all angles 60°. ∠BAC=52°≠60°.

Answer:

No, Yin is not correct. When x=19, the angles are 64°, 64°, and 52°, so the triangle is isosceles but not equilateral.