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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).
Solution :
Given: A triangle $ABC$ where $D$ is the mid-point of side $AB$ (i.e., $AD = DB$). A line $l$ is drawn through $D$ such that $DE \parallel BC$, where $E$ lies on $AC$.
To Prove: $E$ is the mid-point of $AC$ (i.e., $AE = EC$).
Step 1: Stating the Theorem
We use Theorem 6.1, also known as the Basic Proportionality Theorem (BPT). The theorem states: "If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio."
Step 2: Applying the Theorem to Triangle $ABC$
Since $DE \parallel BC$ and $DE$ intersects $AB$ at $D$ and $AC$ at $E$, by the Basic Proportionality Theorem, we have:
$\frac{AD}{DB} = \frac{AE}{EC}$ --- (Equation 1)
Step 3: Utilizing the Given Condition
It is given that $D$ is the mid-point of $AB$. Therefore:
$AD = DB$
Dividing both sides by $DB$, we get:
$\frac{AD}{DB} = 1$ --- (Equation 2)
Step 4: Substituting and Solving
Substitute the value from Equation 2 into Equation 1:
$1 = \frac{AE}{EC}$
Multiplying both sides by $EC$, we obtain:
$EC = AE$
Or, $AE = EC$.
Step 5: Conclusion
Since $AE = EC$, it implies that $E$ is equidistant from $A$ and $C$ on the line segment $AC$. Therefore, $E$ is the mid-point of $AC$.
Final Answer: Hence, it is proved that the line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.
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).
- 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.
- 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 $BC \parallel QR$.
- 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).
- Q9: $ABCD$ is a trapezium in which $AB \parallel DC$ and its diagonals intersect each other at the point $O$. Show that $\frac{AO}{BO} = \frac{CO}{DO}$.
CBSE Solutions for Class 10 Mathematics Triangles
Chapters in CBSE - Class 10 Mathematics
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