The sides of a triangle have the equations y=-1/2x+1, y=2x-4, and y=-3x-9. Verify that the triangle is an isosceles right triangle. Algebraic solutions only!
Method:
solve for the vertices by solving the 3 different pairs of equations.
The take the length of sides between pairs of points using the distance formula √[(x2 - x1)^2 + (y2 - y1)^2]
I will do one of the vertices:
y = 2x-4 with y = -3x-9
2x-4 = -3x-9
5x = -5
x = -1
back in the first ...
y = 2(-1) - 4 = -6
so one vertex is (-1,-6)
let me know if it works out for you
how do i solve the one with the fraction?
do i multiply everything by two?
or put every thing over two?
3(x+2)=12
Actually in terms of method, the method you used for the second point is right
But there was a small error in calculation :)
-(1/2)x + 1 with y = 2x - 4
2x - 4 = -(1/2)x + 1
multiply each term my 2
*accidentally multiplied -4 by 4 to get 16, should be -8
4x - 8 = -x + 2
*And then over here you got 6x= 18 but bringing over that -(1)x would mean 4x +1 x =5x not 6x
5x = 8+2
5x=10
x/5=10/5
x = 2,
*Over here the formula should be y= 2x -4, not y=2x+4 which I'm sure is just a small error you did in a rush
then y=2(2) - 4
y= 4-4
y=0
so another point would be (2,0)
Hope this helped
To determine whether the given triangle is an isosceles right triangle, we need to find the lengths of its sides and check if they satisfy the properties of an isosceles right triangle.
Step 1: Find the coordinates of the points where the three lines intersect.
To find the intersection points of two lines, we need to set their equations equal to each other and solve for the values of x and y.
Setting the first and second line equations equal:
-1/2x + 1 = 2x - 4
Simplifying:
1 = 4.5x - 4
Rearranging:
4.5x = 5
x = 5/4.5
x ≈ 1.11
Substituting the value of x into the second equation to find y:
y = 2(1.11) - 4
y ≈ -1.78
Therefore, the first two lines intersect at the point (1.11, -1.78).
Similarly, setting the first and third line equations equal:
-1/2x + 1 = -3x - 9
Simplifying:
2x - 20x = -10
-18x = -10
x ≈ 0.56
Substituting the value of x into the third equation to find y:
y = -3(0.56) - 9
y ≈ -11.68
Therefore, the first and third lines intersect at the point (0.56, -11.68).
Finally, setting the second and third line equations equal:
2x - 4 = -3x - 9
Simplifying:
5x = -5
x = -1
Substituting the value of x into the second equation to find y:
y = 2(-1) - 4
y ≈ -6
Therefore, the second and third lines intersect at the point (-1, -6).
Step 2: Calculate the lengths of the triangle sides.
Using the distance formula, we can find the lengths of the sides.
For side AB, between points A(1.11, -1.78) and B(0.56, -11.68):
AB = √((x2 - x1)^2 + (y2 - y1)^2)
AB = √((0.56 - 1.11)^2 + (-11.68 + 1.78)^2)
AB ≈ √(0.25 + 103.68)
AB ≈ √103.93
AB ≈ 10.19
For side AC, between points A(1.11, -1.78) and C(-1, -6):
AC = √((x2 - x1)^2 + (y2 - y1)^2)
AC = √((-1 - 1.11)^2 + (-6 + 1.78)^2)
AC ≈ √(2.2121 + 22.2076)
AC ≈ √24.4197
AC ≈ 4.94
For side BC, between points B(0.56, -11.68) and C(-1, -6):
BC = √((x2 - x1)^2 + (y2 - y1)^2)
BC = √((-1 - 0.56)^2 + (-6 + 11.68)^2)
BC ≈ √(2.2276 + 30.4504)
BC ≈ √32.678
BC ≈ 5.71
Step 3: Check if the lengths satisfy the properties of an isosceles right triangle.
An isosceles right triangle has two sides of equal length and one right angle.
In our case, we have AB ≈ 10.19, AC ≈ 4.94, and BC ≈ 5.71.
Since AC is the shortest side and AB and BC are longer and have the same length, we can conclude that the triangle is isosceles.
Moreover, since the square of the longest side BC (approximately 32.678) is almost equal to the sum of the squares of the other two sides (approximately 10.19^2 + 4.94^2 ≈ 156.5119), we can conclude that the triangle is right-angled.
Therefore, the given triangle is an isosceles right triangle.
Ok, I will do one more
y = -(1/2)x + 1 with y = 2x - 4
2x - 4 = -(1/2)x + 1
multiply each term my 2
4x - 16 = -x + 2
6x = 18
x = 3, then y=2(3) + 4 = 10
so another point is (3,10)
now you try the last one, let me know what you got.