Name: Equilateral Triangles
Uploaded: 2019-07-19
Duration: 3 min 15 s
Description: Equilateral triangles have all three sides of equal length. Learn more about such a triangle through this important video having examples.

Question

In the figure below, $O$ is the centre of the circle and $A, B, C$ are points on the circle.

What are the angles of $△ A B C ?$

Accepted Answer

From the given figure,

$A, B, C$

Since sum of a complete whole angle is $360 °$

$\Rightarrow ∠ A O B + ∠ A O C + ∠ B O C = 360 °$
$\Rightarrow 120 ° + 120 ° + ∠ B O C = 360 °$

$\Rightarrow 240 ° + ∠ B O C = 360 °$

$\Rightarrow ∠ B O C = 360 ° - 240 °$ $= 120 °$

In $△ O A B,$
$O A = O B =$ radius of circle
In a triangle with two equal sides, the angles opposite to equal sides are equal.

Let $∠ O A B = ∠ O B A = y °$

And sum of all angles of a triangle is $180 °$

$\Rightarrow ∠ O A B + ∠ O B A + ∠ A O B = 180 °$

$\Rightarrow y ° + y ° + 120 ° = 180 °$

$\Rightarrow 120 ° + 2 y = 180 °$

$\Rightarrow 2 y ° = 180 ° - 120 °$ $= 60 °$

$∴ ∠ O A B = ∠ O B A = 30 ° . . . (1)$

In $△ O A C,$

$O A = O C =$ radius of circle

In a triangle with two equal sides, the angles opposite to equal sides are equal.

And sum of all angles of a triangle is $180 °$

Let $∠ O A C = ∠ O C A = x °$

$\Rightarrow ∠ O A C + ∠ O C A + ∠ A O C = 180 °$

$\Rightarrow x ° + x ° + 120 ° = 180 °$

$\Rightarrow 120 ° + 2 x ° = 180 °$

$\Rightarrow 2 x = 180 ° - 120 ° = 60 °$

$∴ ∠ O A C = ∠ O C A = 30 ° \dots (2)$

In $△ O B C,$
$O B = O C =$ radius of circle
In a triangle with two equal sides, the angles opposite to equal sides are equal.

And sum of all angles of a triangle is $180 °$
$∴ ∠ O C B = ∠ O B C = z °$

$\Rightarrow ∠ O C B + ∠ O B C + ∠ C O B = 180 °$

$\Rightarrow z ° + z ° + 120 ° = 180 °$

$\Rightarrow 120 + 2 z = 180 °$

$\Rightarrow 2 z ° = 180 ° - 120 °$ $= 60 °$

$∴ ∠ O A C = ∠ O B C = 30 ° . . . (3)$

From equations $1, 2$ and $3$ , we get
In $△ A B C$ ,

$∠ A = ∠ B A O + ∠ O A C$ $= 30 ° + 30 °$ $= 60 °$

$∠ B = ∠ O B A + ∠ O B C$ $= 30 ° + 30 ° = 60 °$

And $∠ C = ∠ O C B + ∠ O C A = 30 ° + 30 ° = 60 °$

Thus, the measure of angles of $△ A B C$ is $∠ A = 60 °, ∠ B = 60 °$ and $∠ C = 60 °$

Kerala Board Solutions for Chapter: Equal Triangles, Exercise 5: Exercise 5

Kerala Board Mathematics Solutions for Exercise - Kerala Board Solutions for Chapter: Equal Triangles, Exercise 5: Exercise 5

Questions from Kerala Board Solutions for Chapter: Equal Triangles, Exercise 5: Exercise 5 with Hints & Solutions

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