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प्रश्न
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is isosceles.
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उत्तर
Given that, in two right triangles one side and acute angle of one are equal to the corresponding side and angle of the other
We have to prove that the triangles are congruent
Let us consider two right triangles such that
∠B=∠E=90° .................(1)
AB=DE ..................(2)
∠C=∠F ..................(3)
Now observe the two triangles ABC and DEF
∠C=∠F [From (3)]
∠B=∠E [From (4)]
and AB =DE [From (2)]
So, by AAS congruence criterion, we have
ΔABC≅ΔDEF
∴ The two triangles are congruent
Hence proved
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संबंधित प्रश्न
In ΔABC, AD is the perpendicular bisector of BC (see the given figure). Show that ΔABC is an isosceles triangle in which AB = AC.
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Sides opposite to equal angles of a triangle may be unequal
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Angles opposite to equal sides of a triangle are equal
Which of the following statements are true (T) and which are false (F) :
If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.
Which of the following statements are true (T) and which are false (F):
If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.
In a ΔABC, if ∠B = ∠C = 45°, which is the longest side?
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(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + QB + OC
(iii) OA + OB + OC >` 1/2`(AB + BC + CA)
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Fill in the blank to make the following statement true.
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Fill in the blank to make the following statement true.
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In the given figure, the value of x is ______.
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D is a point on the side BC of a ∆ABC such that AD bisects ∠BAC. Then ______.
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In ∆PQR, ∠P = 70° and ∠R = 30°. Which side of this triangle is the longest? Give reason for your answer.
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