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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

3. Inequalities in One Triangle

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Exercise 53 Page 433

The centroid is the point of concurrency of the medians of a triangle. Use the Concurrency of Medians Theorem.

XK = 9

Practice makes perfect

The centroid is the point of concurrency of the medians of a triangle. In the given diagram, P is the centroid.

Using the Concurrency of Medians Theorem, we can write an equation that we can use to find the desired length XK. XP = 2/3XK With the Segment Addition Postulate, we can rewrite XK as the sum of the two smaller segments. We are given that KP= 3, which we can substitute into this new equation. XK = XP+ KP ⇒ XK = XP+ 3 Next, we can form a system of equations that we can solve using the Substitution Method.

XP = 23XK & (I) XK = XP+3 & (II)

(I): XK= XP+3

XP = 23( XP+3) XK = XP+3

(I): Distribute 2/3

XP = 23XP+2 XK = XP+3

(I): LHS-2/3XP=RHS-2/3XP

13XP = 2 XK = XP+3

(I): LHS * 3=RHS* 3

XP = 6 XK = XP+3

Finally, we can substitute XP=6 into the second equation to find XK.

XP = 6 XK = XP+3

(II): XP= 6

XP = 6 XK = 6+3

(II): Add terms

XP =6 XK = 9

Subchapter links

Exercises p.430-433