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
∴ [tex]\frac{PM}{PB}[/tex] = [tex]\frac{1}{2}[/tex]
∵ PM = 2x + 5
∵ BP = 7x + 4
→ Substitute them in the ratio above
∴ [tex]\frac{2x+5}{7x+4}[/tex] = [tex]\frac{1}{2}[/tex]
→ 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
