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In triangle [tex]\(ABC\)[/tex], segment [tex]\(BQ\)[/tex] is a median and point [tex]\(M\)[/tex] is the centroid. If [tex]\(BM = 9x\)[/tex] and [tex]\(MQ = 5x - 1\)[/tex], what are the values of [tex]\(x\)[/tex] and [tex]\(BQ\)[/tex]?

A. [tex]\(x = 2\)[/tex] and [tex]\(BQ = 27\)[/tex]

B. [tex]\(x = 3\)[/tex] and [tex]\(BQ = 41\)[/tex]

C. [tex]\(x = 3\)[/tex] and [tex]\(BQ = 14\)[/tex]

D. [tex]\(x = 2\)[/tex] and [tex]\(BQ = 18\)[/tex]

Sagot :

GinnyAnswer
To solve the problem, we'll use the fact that in a triangle, the centroid (M) divides each median into a ratio of 2:1, meaning [tex]\( BM = 2 \cdot MQ \)[/tex].

Given the lengths:
[tex]\[ BM = 9x \][/tex]
[tex]\[ MQ = 5x - 1 \][/tex]

We set up the equation from the provided information:
[tex]\[ BM = 2 \cdot MQ \][/tex]
Substitute the given expressions for [tex]\( BM \)[/tex] and [tex]\( MQ \)[/tex]:
[tex]\[ 9x = 2 \cdot (5x - 1) \][/tex]

Now, let's solve this equation step-by-step.

1. Distribute the 2 on the right side:
[tex]\[ 9x = 2 \cdot 5x - 2 \cdot 1 \][/tex]
[tex]\[ 9x = 10x - 2 \][/tex]

2. Bring the [tex]\( 10x \)[/tex] term to the left side by subtracting [tex]\( 10x \)[/tex] from both sides:
[tex]\[ 9x - 10x = -2 \][/tex]
[tex]\[ -x = -2 \][/tex]

3. Multiply both sides by -1 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = 2 \][/tex]

Now that we have [tex]\( x \)[/tex], let's find the length of [tex]\( BQ \)[/tex]. The segment [tex]\( BQ \)[/tex] is the sum of [tex]\( BM \)[/tex] and [tex]\( MQ \)[/tex]:
[tex]\[ BQ = BM + MQ \][/tex]

Substitute the expressions for [tex]\( BM \)[/tex] and [tex]\( MQ \)[/tex]:
[tex]\[ BQ = 9x + (5x - 1) \][/tex]
Now, substitute [tex]\( x = 2 \)[/tex]:
[tex]\[ BQ = 9(2) + 5(2) - 1 \][/tex]
[tex]\[ BQ = 18 + 10 - 1 \][/tex]
[tex]\[ BQ = 27 \][/tex]

Therefore, the correct values are [tex]\( x = 2 \)[/tex] and [tex]\( BQ = 27 \)[/tex].

The correct answer is:
A. [tex]\( x = 2 \)[/tex] and [tex]\( BQ = 27 \)[/tex]
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