Ex.7.2 Q4 Triangles Solution - NCERT Maths Class 9

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\(ABC\) is a triangle in which altitudes \(BE\) and \(CF\) to sides \(AC\) and \(AB\) are equal (see the given figure).

Show that:

(i) \(\ \Delta ABE \cong \Delta ACF\)

(ii) \(AB=AC,\) i.e., ABC is an isosceles triangle.

 Video Solution
Ex 7.2 | Question 4

Text Solution

What is Known?

Altitudes,\(\text{BE}=\text{CF},\text{ BE}\bot \text{AC}\ \text{and CF}\bot \text{AB}\)

To prove:

(i) \(\ \Delta ABE \cong \Delta ACF\)
(ii) \(AB=AC,\) i.e., ABC is an isosceles triangle.


We can show two triangles \(ABE\) and \(ACF\) congruent by using AAS congruency rule and then we can say corresponding parts of congruent triangles will be equal.


(i) In \(\Delta ABE\text{ and }\Delta ACF,\) 

\[\begin{align}& \angle AEB \text{ and }\angle AFC \;(\text{Each} 90^{\circ}) \\  \angle A&=\angle A \;(\text{Common angle}) \\  BE&=CF \; (\text{Given}) \\ \therefore \Delta ABE &\cong \Delta ACF \\&(\text{By AAS congruence rule}) \end{align}\]

(ii) It has already been proved that

\[\begin{align}&\Delta ABE \cong \Delta ACF, \\ &\therefore AB = AC \; (\text{By }CPCT) \\ \end{align}\]

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