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Ex.7.2 Q4 Triangles Solution - NCERT Maths Class 9

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Question

\(ABC\) 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
Triangles
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.

Reasoning:

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.

Steps:

(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

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

 Video Solution
Triangles
Ex 7.2 | Question 4
  
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