Question

Triangles abc, dbe, and fbg are all symmetric about the y-axis. what are the coordinates of the centroid?

please help

Answer 1

The coordinates of the centroid are (0, 5)

Explanation:

The coordinates of the centroid of a triangle whose vertices (x1, y1), (x2, y2), and (x3, y3) are (
`(x1+x2+x3)/(3)` ,
`(y1+y2+y3)/(3)`)

In Δ ABC

∵ A = (-20, 0), B = (0, 15), C = (20, 0)

∴ x1 = -20, x2 = 0, x3 = 20

∴ y1 = 0, y2 = 15, y3 = 0

∴ The centroid = (
`(-20+0+20)/(3)` ,
`(0+15+0)/(3)`) = (
`(0)/(3)` ,
`(15)/(3)`) = (0, 5)

∴ The coordinates of the centroid of Δ ABC are (0, 5)

In Δ DBE

∵ D = (-15, 0), B = (0, 15), E = (15, 0)

∴ x1 = -15, x2 = 0, x3 = 15

∴ y1 = 0, y2 = 15, y3 = 0

∴ The centroid = (
`(-15+0+15)/(3)` ,
`(0+15+0)/(3)`) = (
`(0)/(3)` ,
`(15)/(3)`) = (0, 5)

∴ The coordinates of the centroid of Δ DBE are (0, 5)

In Δ FBG

∵ F = (-5, 0), B = (0, 15), G = (5, 0)

∴ x1 = -5, x2 = 0, x3 = 5

∴ y1 = 0, y2 = 15, y3 = 0

∴ The centroid = (
`(-5+0+5)/(3)` ,
`(0+15+0)/(3)`) = (
`(0)/(3)` ,
`(15)/(3)`) = (0, 5)

∴ The coordinates of the centroid of Δ FBG are (0, 5)

∵ The three triangles are symmetric about the y-axis

→ That means they have the same centroid and it lies on the y-axis

∴ The coordinates of the centroid are (0, 5)