Advertisements
Advertisements
प्रश्न
In a triangle ABC, D is the mid-point of side AC such that BD = `1/2` AC. Show that ∠ABC is a right angle.
Advertisements
उत्तर
Given: In ΔABC, D is the mid-point of AC i.e., AD = CD such that BD = `1/2` AC.
To show: ∠ABC = 90°
Proof: We have BD = `1/2` AC ...(i)
Since, D is the mid-point of AC.
∴ AD = CD = `1/2` AC ...(ii)
From equations (i) and (ii),
AD = CD = BD
In ΔDAB, AD = BD ...[Proved above]
∴ ∠ABD = ∠BAD ...(iii) [Angles opposite to equal sides are equal]
In ΔDBC, BD = CD ...[Proved above]
∴ ∠BCD = ∠CBD ...(iv) [Angles opposite to equal sides are equal]
In ΔABC, ∠ABC + ∠BAC + ∠ACB = 180° ...[By angle sum property of a triangle]
⇒ ∠ABC + ∠BAD + ∠DCB = 180°
⇒ ∠ABC + ∠ABD + ∠CBD = 180° ...[From equations (iii) and (iv)]
⇒ ∠ABC + ∠ABC = 180°
⇒ 2∠ABC = 180°
⇒ ∠ABC = 90°
APPEARS IN
संबंधित प्रश्न
In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:
- OB = OC
- AO bisects ∠A
Prove that the medians of an equilateral triangle are equal.
In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
P is a point on the bisector of an angle ∠ABC. If the line through P parallel to AB meets BC at Q, prove that triangle BPQ is isosceles.
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) :
If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.
Fill the blank in the following so that the following statement is true.
Sides opposite to equal angles of a triangle are ......
Which of the following statements are true (T) and which are false (F)?
Sum of any two sides of a triangle is greater than the third side.
Fill in the blank to make the following statement true.
The sum 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 || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
Which of the following correctly describes the given triangle?
In ∆PQR, ∠R = ∠P and QR = 4 cm and PR = 5 cm. Then the length of PQ is ______.
It is given that ∆ABC ≅ ∆FDE and AB = 5 cm, ∠B = 40° and ∠A = 80°. Then which of the following is true?
If ∆PQR ≅ ∆EDF, then is it true to say that PR = EF? Give reason for your answer
In ∆PQR, ∠P = 70° and ∠R = 30°. Which side of this triangle is the longest? 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
