In an isosceles triangle ABC with equal sides AC and CB and an apex angle C equal to 120 degrees

| June 27, 2021 | education

In an isosceles triangle ABC with equal sides AC and CB and an apex angle C equal to 120 degrees, bisectors AM and BN are drawn. Find the length of the bisector BN if the area of ANMB is 12.25.

Since triangle ABC is isosceles, its angles at the base of AB are equal.

Angle ABC = BAC = (180 – 120) / 2 = 30.

Since BN and AM are bisectors of the same angles, then the angle BAM = ABN = 30/2 = 150, then the angle AOB = (180 – 15 – 15) = 150.

The quadrilateral BMNA is an isosceles trapezoid, and AM and BN are its diagonals.

The PTO angle is adjacent to the AOB angle, then the BOM angle between the trapezoid diagonals is: 180 – 150 = 30.

We use the trapezoid formula through the diagonals and the angle between them.

S = AM ^ 2 * SinBOM / 2.

Then AM ^ 2 = 2 * S / Sin30 = 2 * 12.25 / (1/2) = 4 * 12.25 = 49.

AM = 7 cm.

Answer: The length of the bisector is 7 cm.



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