Advertisements
Advertisements
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.
Advertisements
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
APPEARS IN
संबंधित प्रश्न
Show that the angles of an equilateral triangle are 60° each.
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ΔABC is isosceles
Which of the following statements are true (T) and which are false (F):
The bisectors of two equal angles of a triangle are equal
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 ΔABC, side AB is produced to D so that BD = BC. If ∠B = 60° and ∠A = 70°, prove that: (i) AD > CD (ii) AD > AC
Which of the following statements are true (T) and which are false (F)?
Sum of the three sides of a triangle is less than the sum of its three altitudes.
Fill in the blank to make the following statement true.
The sum of three altitudes of a triangle is ..... than its perimeter.
Fill in the blank to make the following statement true.
Difference of any two sides of a triangle is........ than the third side.
Fill in the blank to make the following statement true.
If two sides of a triangle are unequal, then the larger side has .... angle opposite to it.
In the given figure, if AB ⊥ BC. then x =
If the bisectors of the acute angles of a right triangle meet at O, then the angle at Obetween the two bisectors is
The angles of a right angled triangle are
In ∆ABC, AB = AC and ∠B = 50°. Then ∠C is equal to ______.
If ∆PQR ≅ ∆EDF, then is it true to say that PR = EF? Give reason for your answer
Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC
