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Q8:
Using Theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side. (Recall that you have done it in Class IX).
Solution :
Given: A triangle $ABC$ where $D$ is the mid-point of side $AB$ and $E$ is the mid-point of side $AC$.
To Prove: The line segment $DE$ is parallel to the third side $BC$ (i.e., $DE \parallel BC$).
Theorem Used: Theorem 6.2 (Converse of Basic Proportionality Theorem): If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
Step 1: Establishing the ratios based on mid-point definitions.
Since $D$ is the mid-point of $AB$, by definition:
$AD = DB$
Dividing both sides by $DB$, we get:
$\frac{AD}{DB} = 1$ --- (Equation 1)
Step 2: Establishing the ratio for the second side.
Since $E$ is the mid-point of $AC$, by definition:
$AE = EC$
Dividing both sides by $EC$, we get:
$\frac{AE}{EC} = 1$ --- (Equation 2)
Step 3: Comparing the ratios.
From Equation 1 and Equation 2, we observe that both ratios are equal to 1:
$\frac{AD}{DB} = \frac{AE}{EC} = 1$
[Since both ratios are equal to the same value, they are equal to each other.]
Step 4: Applying the Converse of the Basic Proportionality Theorem.
According to Theorem 6.2, if a line $DE$ intersects sides $AB$ and $AC$ of $\triangle ABC$ such that $\frac{AD}{DB} = \frac{AE}{EC}$, then the line $DE$ must be parallel to the third side $BC$.
Therefore, $DE \parallel BC$.
Conclusion:
We have shown that the line segment joining the mid-points of two sides of a triangle divides those sides in the same ratio, which necessitates that the segment is parallel to the third side.
Final Answer: Hence, it is proved that $DE \parallel BC$.
More Questions from Class 10 Mathematics Triangles EXERCISE 6.2
- Q1(i): In Fig. 6.17, (i) and (ii), $DE \parallel BC$. Find $EC$ in (i).
- Q1(ii): In Fig. 6.17, (i) and (ii), $DE \parallel BC$. Find $AD$ in (ii).
- Q2(i): $E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\triangle PQR$. For each of the following cases, state whether $EF \parallel QR$ : (i) $PE = 3.9$ cm, $EQ = 3$ cm, $PF = 3.6$ cm and $FR = 2.4$ cm
- Q2(ii): $E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\triangle PQR$. For each of the following cases, state whether $EF \parallel QR$ : (ii) $PE = 4$ cm, $QE = 4.5$ cm, $PF = 8$ cm and $RF = 9$ cm
- Q2(iii): $E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\triangle PQR$. For each of the following cases, state whether $EF \parallel QR$ : (iii) $PQ = 1.28$ cm, $PR = 2.56$ cm, $PE = 0.18$ cm and $PF = 0.36$ cm
- Q3: In Fig. 6.18, if $LM \parallel CB$ and $LN \parallel CD$, prove that $\frac{AM}{AB} = \frac{AN}{AD}$.
- Q4: In Fig. 6.19, $DE \parallel AC$ and $DF \parallel AE$. Prove that $\frac{BF}{FE} = \frac{BE}{EC}$.
- Q5: In Fig. 6.20, $DE \parallel OQ$ and $DF \parallel OR$. Show that $EF \parallel QR$.
- Q6: In Fig. 6.21, $A$, $B$ and $C$ are points on $OP$, $OQ$ and $OR$ respectively such that $AB \parallel PQ$ and $AC \parallel PR$. Show that $BC \parallel QR$.
- Q7: Using Theorem 6.1, prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. (Recall that you have proved it in Class IX).
- Q9: $ABCD$ is a trapezium in which $AB \parallel DC$ and its diagonals intersect each other at the point $O$. Show that $\frac{AO}{BO} = \frac{CO}{DO}$.
- Q10: The diagonals of a quadrilateral $ABCD$ intersect each other at the point $O$ such that $\frac{AO}{BO} = \frac{CO}{DO}$. Show that $ABCD$ is a trapezium.
CBSE Solutions for Class 10 Mathematics Triangles
Chapters in CBSE - Class 10 Mathematics
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