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If the vertices of the triangle are A(1,2), B(-2,-3) and C(2,3) then which is the largest angle?
- a)Angle(ABC)
- b)Angle(BAC)
- c)Angle(ACB)
- d)None
Correct answer is option 'B'. Can you explain this answer?
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If the vertices of the triangle are A(1,2), B(-2,-3) and C(2,3) then w...
To determine which angle is the largest in triangle ABC, we can use the concept of slopes and the distance formula.
Step 1: Find the lengths of the sides of the triangle
Using the distance formula, we can calculate the lengths of the sides of the triangle ABC.
Side AB:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(-2 - 1)^2 + (-3 - 2)^2]
= √[(-3)^2 + (-5)^2]
= √[9 + 25]
= √34
Side BC:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(2 - (-2))^2 + (3 - (-3))^2]
= √[(2 + 2)^2 + (3 + 3)^2]
= √[4^2 + 6^2]
= √[16 + 36]
= √52
= 2√13
Side AC:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(2 - 1)^2 + (3 - 2)^2]
= √[(1)^2 + (1)^2]
= √[1 + 1]
= √2
Step 2: Find the slopes of the sides of the triangle
Using the slope formula, we can calculate the slopes of the sides of triangle ABC.
Slope of AB:
m = (y2 - y1) / (x2 - x1)
= (-3 - 2) / (-2 - 1)
= (-5) / (-3)
= 5/3
Slope of BC:
m = (y2 - y1) / (x2 - x1)
= (3 - (-3)) / (2 - (-2))
= (3 + 3) / (2 + 2)
= 6/4
= 3/2
Slope of AC:
m = (y2 - y1) / (x2 - x1)
= (3 - 2) / (2 - 1)
= (1) / (1)
= 1
Step 3: Find the angles of the triangle
Using the slopes, we can find the angles of the triangle using the arctan function.
Angle ABC: tan(ABC) = |(m2 - m1) / (1 + (m1 * m2))|
= |((3/2) - (5/3)) / (1 + ((5/3) * (3/2)))|
= |(9/6 - 10/6) / (1 + (15/6))|
= |(-1/6) / (6/6 + 15/6)|
= |-1/6| / (21/6)
= 1/6 / (7/2)
= 1/6 * 2/7
=
Step 1: Find the lengths of the sides of the triangle
Using the distance formula, we can calculate the lengths of the sides of the triangle ABC.
Side AB:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(-2 - 1)^2 + (-3 - 2)^2]
= √[(-3)^2 + (-5)^2]
= √[9 + 25]
= √34
Side BC:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(2 - (-2))^2 + (3 - (-3))^2]
= √[(2 + 2)^2 + (3 + 3)^2]
= √[4^2 + 6^2]
= √[16 + 36]
= √52
= 2√13
Side AC:
Length = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(2 - 1)^2 + (3 - 2)^2]
= √[(1)^2 + (1)^2]
= √[1 + 1]
= √2
Step 2: Find the slopes of the sides of the triangle
Using the slope formula, we can calculate the slopes of the sides of triangle ABC.
Slope of AB:
m = (y2 - y1) / (x2 - x1)
= (-3 - 2) / (-2 - 1)
= (-5) / (-3)
= 5/3
Slope of BC:
m = (y2 - y1) / (x2 - x1)
= (3 - (-3)) / (2 - (-2))
= (3 + 3) / (2 + 2)
= 6/4
= 3/2
Slope of AC:
m = (y2 - y1) / (x2 - x1)
= (3 - 2) / (2 - 1)
= (1) / (1)
= 1
Step 3: Find the angles of the triangle
Using the slopes, we can find the angles of the triangle using the arctan function.
Angle ABC: tan(ABC) = |(m2 - m1) / (1 + (m1 * m2))|
= |((3/2) - (5/3)) / (1 + ((5/3) * (3/2)))|
= |(9/6 - 10/6) / (1 + (15/6))|
= |(-1/6) / (6/6 + 15/6)|
= |-1/6| / (21/6)
= 1/6 / (7/2)
= 1/6 * 2/7
=
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