Question
As shown in the given figure, a student drew ∆ABC using five parallel lines of a notebook. Then he found the centroid G of the triangle. How will you decide whether the location of G he found, is correct.
✓
Answer
Draw seg AP ⊥ seg PE and seg EQ ⊥ seg QC.
Side AP || side EQ and AC is their transversal.
∴ ∠PAE ≅ ∠QEC …(i) [Corresponding angles]
In ∆ APE and ∆ EQC,
∠PAE ≅ ∠QEC …[From (i)]
∠APE ≅ ∠EQC
… [Each angle is of measure 90°]
side PE ≅ side QC
…. [Perpendicular distance between parallel lines]
∴ ∆ APE ≅ ∆ EQC … [By AAS test]
∴ AE = EC
… [Corresponding sides of congruent triangles]
∴ E is the midpoint of AC.
∴ seg BE is the median.
Similarly, seg CF is the median.
Since, the medians of a triangle are concurrent.
∴ G is the centroid of ∆ABC.
Side AP || side EQ and AC is their transversal.
∴ ∠PAE ≅ ∠QEC …(i) [Corresponding angles]
In ∆ APE and ∆ EQC,
∠PAE ≅ ∠QEC …[From (i)]
∠APE ≅ ∠EQC
… [Each angle is of measure 90°]
side PE ≅ side QC
…. [Perpendicular distance between parallel lines]
∴ ∆ APE ≅ ∆ EQC … [By AAS test]
∴ AE = EC
… [Corresponding sides of congruent triangles]
∴ E is the midpoint of AC.
∴ seg BE is the median.
Similarly, seg CF is the median.
Since, the medians of a triangle are concurrent.
∴ G is the centroid of ∆ABC.
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Draw medians; seg AR, seg BQ and seg CP of ∆ABC.
Name the point of concurrence as G.
Measure the lengths of segments from the figure and fill in the boxes in the following table.
\begin{tabular}{|l|l|l|}
\hline$I(A G)=$ & $I(G R)=$ & $I(A G): I(G R)=$ \\
\hline$I(B G)=$ & $I(G Q)=$ & $I(B G): I(G Q)=$ \\
\hline$I(C G)=$ & $I(G P)=$ & $I(C G): I(G P)=$ \\
\hline
\end{tabular}
→Draw medians; seg AR, seg BQ and seg CP of ∆ABC.
Name the point of concurrence as G.
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\begin{tabular}{|l|l|l|}
\hline$I(A G)=$ & $I(G R)=$ & $I(A G): I(G R)=$ \\
\hline$I(B G)=$ & $I(G Q)=$ & $I(B G): I(G Q)=$ \\
\hline$I(C G)=$ & $I(G P)=$ & $I(C G): I(G P)=$ \\
\hline
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