Question

Find an equation of the line of symmetry of triangle ABC.

Answer 1

The equation of the line of symmetry of triangle ABC is:

• 3x + 2y = 86

Explanation:

The line of symmetry of an isosceles triangle bisects the vertex angle (the angle opposite the base) and is perpendicular to the base of the triangle.

As AB = AC, then angle A is the vertex angle of the isosceles triangle ABC.

If B and C lie on the line with equation 3y = 2x + 12, then the line of symmetry is the line that is perpendicular to this line and passes through A (4, 37).

Rearrange the given equation 3y = 2x + 12 to slope-intercept form by dividing both sides by 3:

`\implies y=(2)/(3)x+4`

Perpendicular lines have slopes that are negative reciprocals of one another. Therefore, the slope of the perpendicular line is -³/₂.

Substitute the found slope and point A (4, 37) into the point-slope formula:

`\implies y-y_1=m(x-x_1)`

`\implies y-37=-(3)/(2)(x-4)`

Rearrange the equation to standard form:

`\implies 2y-74=-3(x-4)`

`\implies 2y-74=-3x+12`

`\implies 3x+2y-74=12`

`\implies 3x+2y=86`

Therefore, the equation of the line of symmetry of triangle ABC is:

• 3x + 2y = 86