Question

In the diagram below, point P is the centroid of △ABC. If PM=2x+5 and BP= 7x+4, what is the length of PM?​

Answer 1

Answer:

The length of PM is 9 ⇒ (1)

Step-by-step explanation:

• The median of a triangle is the line that drawn from a vertex to the mid-point of its opposite side

• The centroid of a triangle is the point of intersection of its three medians

• The centroid of a triangle divides each median at a ratio 1: 2 from the base

In Δ ABC

∵ Point P is the centroid of the triangle ABC

∵ BM passes through point P

∴ BM is a median

→ By using the 3rd note above

∴ PM : PB = 1 : 2

∴ $\frac{PM}{PB}$ = $\frac{1}{2}$

∵ PM = 2x + 5

∵ BP = 7x + 4

→ Substitute them in the ratio above

∴ $\frac{2x+5}{7x+4}$ =  $\frac{1}{2}$

→ By using cross multiplication

∵ (7x + 4) × 1 = 2 × (2x + 5)

∴ 7x + 4 = 2(2x) + 2(5)

∴ 7x + 4 = 4x + 10

→ Subtract 4x from both sides

∵ 7x - 4x + 4 = 4x - 4x + 10

∴ 3x + 4 = 10

→ Subtract 4 from both sides

∵ 3x + 4 - 4 = 10 - 4

∴ 3x = 6

→ Divide both sides by 3

∴ x = 2

→ Substitute the value of x in the expression of PM

∵ PM = 2(2) + 5

∴ PM = 4 + 5

∴ PM = 9

∴ The length of PM is 9