Question:
If a line intersects sides \( AB \) and \( AC \) of a triangle \( ABC \) at \( D \) and \( E \) respectively and is parallel to \( BC \), prove that \( \frac{AD}{AB} = \frac{AE}{AC} \).
If a line intersects sides \( AB \) and \( AC \) of a triangle \( ABC \) at \( D \) and \( E \) respectively and is parallel to \( BC \), prove that \( \frac{AD}{AB} = \frac{AE}{AC} \).
Show Hint
In a triangle, if a line is parallel to one side, it divides the other two sides in the same ratio. This is known as the Basic Proportionality Theorem.
Updated On: Oct 10, 2025
Hide Solution
Verified By Collegedunia
Solution and Explanation
We are given a triangle \( ABC \) and a line passing through points \( D \) and \( E \) on sides \( AB \) and \( AC \), respectively, such that the line is parallel to \( BC \).
By the Basic Proportionality Theorem (also known as Thales' Theorem), if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those two sides in the same ratio. Therefore, we have:
\[
\frac{AD}{AB} = \frac{AE}{AC}.
\]
Thus, the required result is proved.
Conclusion: \[ \frac{AD}{AB} = \frac{AE}{AC}. \]
Conclusion: \[ \frac{AD}{AB} = \frac{AE}{AC}. \]
Was this answer helpful?
0
0
Top Questions on Triangles
- If the orthocentre of the triangle formed by the lines $ y = x + 1 $, $ y = 4x - 8 $, and $ y = mx + c $ is at $ (3, -1) $, then $ m - c $ is:
In the adjoining figure, \( AP = 1 \, \text{cm}, \ BP = 2 \, \text{cm}, \ AQ = 1.5 \, \text{cm}, \ AC = 4.5 \, \text{cm} \) Prove that \( \triangle APQ \sim \triangle ABC \).
Hence, find the length of \( PQ \), if \( BC = 3.6 \, \text{cm} \).
- ABC is an isosceles right triangle with C as right angle. Prove that \(AB^2 = 2AC^2\).
- In \(\triangle DEF\) and \(\triangle PQR\) it is given that \(\angle D = \angle Q\) and \(\angle R = \angle E\), then which of the following is correct?
- In any \(\triangle ABC\), \(\angle A = 90^\circ\), BC = 13 cm, AB = 12 cm; then the value of AC is
View More Questions
Questions Asked in UP Board X exam
- The altitude of a right-angled triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
- UP Board X - 2025
- Right-Angled Triangles And Pythagoras Property
- If the sum of first 7 terms of an A.P. is 49 and the sum of first 17 terms is 289, find the sum of first $n$ terms.
- UP Board X - 2025
- Sum of First n Terms of an AP
Find the unknown frequency if 24 is the median of the following frequency distribution:
\[\begin{array}{|c|c|c|c|c|c|} \hline \text{Class-interval} & 0-10 & 10-20 & 20-30 & 30-40 & 40-50 \\ \hline \text{Frequency} & 5 & 25 & 25 & \text{$p$} & 7 \\ \hline \end{array}\]- UP Board X - 2025
- Median of Grouped Data
- If the roots of the equation $x^2+2(m-1)x+(m+5)=0$ are real and equal, find the value of $m$.
- UP Board X - 2025
- Quadratic Equations
Two concentric circles are of radii $8\ \text{cm}$ and $5\ \text{cm}$. Find the length of the chord of the larger circle which touches (is tangent to) the smaller circle.
- UP Board X - 2025
- Circles, Chords and Tangents
View More Questions