A,B & C are the points (-8, -4), (-2, 6) and (4, -4) respectively. show that ABC is an isosceles triangle. find the perimeter of the triangle
xA = - 8 , yA = - 4
xB = - 2 , yB = 6
xC = 4 , yC = - 4
AB = √ ( ( xA - xB)² + ( yA - yB )² )
AB = √ ( ( - 8 - ( - 2) )² + ( - 4 - 6 )² )
AB = √ ( ( - 8 + 2)² + ( - 10 )² )
AB = √ ( ( - 6 )² + 100 )
AB = √ ( 36 + 100 )
AB = √ 136
AB = √ ( 4 ∙ 34 )
AB = √4 ∙ √34
AB = 2 √34
BC = √ ( ( xB - xC)² + ( yB - yC )² )
BC = √ ( ( - 2 - 4 )² + ( 6 - ( - 4 )² )
BC = √ ( ( - 6 )² + ( 6 + 4 )² )
BC = √ ( 36 + 10² )
BC = √ ( 36 + 100 )
BC= √ 136
BC = √ ( 4 ∙ 34 )
BC = √4 ∙ √34
BC = 2 √34
AC = √ ( ( xA - xC)² + ( yA - yC )² )
AC = √ ( ( - 8 - 4)² + ( - 4 - (- 4 ) )² )
AC = √ ( ( - 12 )² + ( - 4 + 4 )² )
AC = √ ( 144 + 0² )
AC = √ 144
AC = √12
Your triangle has two sides of equal length AB = 2 √34 and BC = 2 √34
So your triangle is isosceles triangle.
Perimeter:
P = AB + BC + AC
P = 2 √34 + 2 √34 + √12
P = 4 √34 + 12
P = 4 √34 + 4 ∙ 3
P = 4 ( √34 + 3 )
Well, ABC being an isosceles triangle means that at least two of its sides are of equal length. To prove this, we can calculate the distances between the three points.
Using the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
Distance between A (-8, -4) and B (-2, 6):
dAB = √((-2 - (-8))^2 + (6 - (-4))^2)
= √((6)^2 + (10)^2)
= √(36 + 100)
= √136
= 2√34
Distance between B (-2, 6) and C (4, -4):
dBC = √((4 - (-2))^2 + (-4 - 6)^2)
= √((6)^2 + (-10)^2)
= √(36 + 100)
= √136
= 2√34
Distance between C (4, -4) and A (-8, -4):
dCA = √((-8 - 4)^2 + (-4 - (-4))^2)
= √((-12)^2 + (0)^2)
= √(144 + 0)
= √144
= 12
As we can see, dAB = dBC = 2√34, while dCA = 12. Since two of the distances are equal, ABC is an isosceles triangle.
To find the perimeter of the triangle, we add up the lengths of all three sides:
Perimeter = dAB + dBC + dCA
= 2√34 + 2√34 + 12
= 4√34 + 12
So the perimeter of the triangle ABC is 4√34 + 12!
To show that triangle ABC is an isosceles triangle, we need to prove that two sides of the triangle are equal in length.
Let's find the lengths of the three sides of the triangle:
1. Side AB:
Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((-2 - (-8))^2 + (6 - (-4))^2)
d = sqrt(6^2 + 10^2)
d = sqrt(36 + 100)
d = sqrt(136)
2. Side AC:
Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((4 - (-8))^2 + (-4 - (-4))^2)
d = sqrt(12^2 + 0^2)
d = sqrt(144 + 0)
d = sqrt(144)
3. Side BC:
Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((4 - (-2))^2 + (-4 - 6)^2)
d = sqrt(6^2 + (-10)^2)
d = sqrt(36 + 100)
d = sqrt(136)
Based on the lengths calculated:
AB = sqrt(136)
AC = sqrt(144)
BC = sqrt(136)
We can see that AB = BC. Therefore, the triangle ABC is isosceles.
To find the perimeter of the triangle, we need to calculate the sum of all three sides:
Perimeter = AB + AC + BC
= sqrt(136) + sqrt(144) + sqrt(136)
Now, we can calculate the sum to find the perimeter.
To show that ABC is an isosceles triangle, we need to prove that at least two sides of the triangle are congruent.
To find the distances between the points A, B, and C, we can use the distance formula:
The distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2-x1)^2 + (y2-y1)^2)
Using this formula, we can find the distances between the points:
Distance between A and B (AB):
AB = sqrt((-2 - (-8))^2 + (6 - (-4))^2)
= sqrt((6)^2 + (10)^2)
= sqrt(36 + 100)
= sqrt(136)
Distance between B and C (BC):
BC = sqrt((4 - (-2))^2 + (-4 - 6)^2)
= sqrt((6)^2 + (-10)^2)
= sqrt(36 + 100)
= sqrt(136)
Distance between A and C (AC):
AC = sqrt((4 - (-8))^2 + (-4 - (-4))^2)
= sqrt((12)^2 + (0)^2)
= sqrt(144 + 0)
= sqrt(144)
= 12
Now, we can compare the distances to determine if ABC is an isosceles triangle:
AB = sqrt(136)
BC = sqrt(136)
AC = 12
Since AB = BC, we have proven that at least two sides of the triangle are congruent. Therefore, ABC is an isosceles triangle.
To find the perimeter of the triangle, we just need to add up the lengths of all three sides:
Perimeter (P) = AB + BC + AC
P = sqrt(136) + sqrt(136) + 12
P = 2 * sqrt(136) + 12
Therefore, the perimeter of triangle ABC is 2 * sqrt(136) + 12 units.