Advertisements
Advertisements
प्रश्न
M is a point on side BC of a triangle ABC such that AM is the bisector of ∠BAC. Is it true to say that perimeter of the triangle is greater than 2 AM? Give reason for your answer.
Advertisements
उत्तर
Yes, In ΔABC, M is a point of side BC such that AM is the bisector of ∠BAC.
In ΔABM, AB + BM > AM ...(i) [Sum of two sides of a triangle is greater than the third side]
In ΔACM, AC + CM > AM ...(ii) [Sum of two sides of a triangle is greater than the third side]
On adding equations (i) and (ii), we get
(AB + BM + AC + CM) > 2AM
⇒ (AB + BM + MC + AC) > 2AM
⇒ AB + BC + AC > 2AM ...[∵ BC = BM + MC]
∴ Perimeter of ΔABC > 2AM
APPEARS IN
संबंधित प्रश्न
ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.
Prove that the medians of an equilateral triangle are equal.
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
Prove that each angle of an equilateral triangle is 60°.
In a ΔPQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP
respectively. Prove that LN = MN.
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 measure of each angle of an equilateral triangle is 60°
Fill the blank in the following so that the following statement is true.
In a ΔABC if ∠A = ∠C , then AB = ......
Fill the blank in the following so that the following statement is true.
If altitudes CE and BF of a triangle ABC are equal, then AB = ....
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
O is any point in the interior of ΔABC. Prove that
(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + QB + OC
(iii) OA + OB + OC >` 1/2`(AB + BC + CA)
Which of the following statements are true (T) and which are false (F)?
Difference of any two sides of a triangle is equal to the third side.
In a ΔABC, if ∠A = 60°, ∠B = 80° and the bisectors of ∠B and ∠C meet at O, then ∠BOC =
In a ΔABC, ∠A = 50° and BC is produced to a point D. If the bisectors of ∠ABC and ∠ACDmeet at E, then ∠E =
In ∆PQR, if ∠R > ∠Q, then ______.
In triangles ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are ______.
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
