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Q1(i):
In Fig. 6.17, (i) and (ii), $DE \parallel BC$. Find $EC$ in (i).
In Fig. 6.17, (i) and (ii), $DE \parallel BC$. Find $EC$ in (i).
Solution :
Given: In $\triangle ABC$, $DE \parallel BC$. The lengths of the segments are provided as follows: $AD = 1.5\text{ cm}$, $DB = 3\text{ cm}$, and $AE = 1\text{ cm}$.
To find: The length of segment $EC$.
Step 1: Identifying the Applicable Theorem
Since $DE \parallel BC$ in $\triangle ABC$, we apply the Basic Proportionality Theorem (Thales Theorem). The theorem states that 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.
Mathematically, this is expressed as: $$\frac{AD}{DB} = \frac{AE}{EC}$$ [By Basic Proportionality Theorem]
Step 2: Substituting the Given Values
We substitute the known values into the equation:
$$\frac{1.5}{3} = \frac{1}{EC}$$
Step 3: Solving for $EC$
To isolate $EC$, we perform cross-multiplication:
$$1.5 \times EC = 3 \times 1$$
$$1.5 \times EC = 3$$
Now, divide both sides by $1.5$: $$EC = \frac{3}{1.5}$$
To simplify the division, multiply the numerator and denominator by 10: $$EC = \frac{30}{15}$$ $$EC = 2$$
Step 4: Conclusion
The length of segment $EC$ is calculated to be $2\text{ cm}$.
Final Answer: $EC = 2\text{ cm}$
More Questions from Class 10 Mathematics Triangles EXERCISE 6.2
- 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 $BC \parallel QR$.
- 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).
- 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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