Question

round to the nearest tenth if necessary
find dp.
answer attempt 4 out of 99
x =
ad =
ap =
dp =

Explanation:

Step1: Identify Median Property

In a triangle, the centroid divides each median into a ratio of \(2:1\) (the segment from vertex to centroid is twice the segment from centroid to midpoint). Also, \(D\) and \(E\) are midpoints (since \(P\) is centroid), so \(AD = DB\). Thus, \(2x + 14=8x - 52\).

Step2: Solve for \(x\)

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Step3: Find \(AD\)

Substitute \(x = 11\) into \(AD = 2x + 14\):
\(AD = 2(11)+14 = 22 + 14 = 36\).

Step4: Find \(DP\)

Since centroid divides median \(AB\) (where \(AD = DB\)) into \(AP:PD = 2:1\), let \(DP = y\), then \(AP = 2y\), and \(AD=AP + PD=3y\).
We know \(AD = 36\), so \(3y = 36\Rightarrow y = 12\). Alternatively, since \(AD = 36\) and \(AP:PD = 2:1\), \(DP=\frac{1}{3}AD=\frac{36}{3}=12\). (Also, check centroid property with other medians, but here using \(AD\) is sufficient as \(D\) is midpoint.)

Answer:

\(x = \boldsymbol{11}\)
\(AD = \boldsymbol{36}\)
\(AP = \boldsymbol{24}\) (since \(AP = 2\times12\))
\(DP = \boldsymbol{12}\)