Mathematics
The perpendicular bisectors of the sides of a triangle ABC meet at I.
Prove that : IA = IB = IC.
Triangles
26 Likes
Answer
From figure,
AD, BE and CF are perpendicular bisectors of sides BC, AC and AB, respectively.
In △ BID and △ CID,
⇒ BD = CD (Since, AD is the perpendicular bisector of BC)
⇒ ∠BDI = ∠CDI (Both equal to 90°)
⇒ DI = DI (Common side)
∴ △ BID ≅ △ CID (By S.A.S. axiom)
We know that,
Corresponding parts of congruent triangles are equal.
By C.P.C.T.C.,
⇒ IB = IC ……..(1)
In △ CIE and △ AIE,
⇒ CE = AE (Since, BE is the perpendicular bisector of AC)
⇒ ∠CEI = ∠AEI (Both equal to 90°)
⇒ IE = IE (Common side)
∴ △ CIE ≅ △ AIE (By S.A.S. axiom)
Since, △ CIE and △ AIE are congruent.
By C.P.C.T.C.,
⇒ IC = IA ………(2)
From equation (1) and (2), we get :
⇒ IA = IB = IC.
Hence, proved that IA = IB = IC.
Answered By
17 Likes
Related Questions
In the following figure, AB = EF, BC = DE and ∠B = ∠E = 90°. Prove that AD = FC.
A point O is taken inside a rhombus ABCD such that its distances from the vertices B and D are equal. Show that AOC is a straight line.
In the given figure, ABCD is a rectangle and X and Y are the mid-points of the sides DC and AB respectively. P and Q are the points of AD and BC respectively such that DP = BQ.
Prove that △ APX ≅ △ CQY.
![In the given figure, ABCD is a rectangle and X and Y are the mid-points of the sides DC and AB respectively. P and Q are the points of AD and BC respectively such that DP = BQ. Triangles [Congruency in Triangles], Concise Mathematics Solutions ICSE Class 9.](https://cdn1.knowledgeboat.com/img/cm9/q12-test-yourself-c9-icse-class-9-concise-maths-upd-2027-new-updated-658x429.png)
Mrs. Sharmila, a mathematics teacher of class 9 started a very important geometry chapter on Congruency in triangles and decided to test the previous concept by setting a geometry problem on the blackboard. In △ ABC, AD is drawn as perpendicular bisector of BC. It is given that AD = 9 cm and BC = 24 cm.
![Mrs. Sharmila, a mathematics teacher of class 9 started a very important geometry chapter on Congruency in triangles and decided to test the previous concept by setting a geometry problem on the blackboard. In △ ABC, AD is drawn as perpendicular bisector of BC. It is given that AD = 9 cm and BC = 24 cm. Triangles [Congruency in Triangles], Concise Mathematics Solutions ICSE Class 9.](https://cdn1.knowledgeboat.com/img/cm9/q1-case-study-c9-icse-class-9-concise-maths-upd-2027-1200x769.png)
Based on the above information answer the following:
(i) Prove △ ABD ≅ △ ACD
(ii) Assign a special name to △ ABC
(iii) Calculate perimeter of △ABC.
(iv) Calculate Area of △ABC.