The sides of the triangle at right are formed by the graphs of 3x+ 2y = 1, y = x − 2, and −4x+ 9y = 22. Is the triangle isosceles? How do you know?
A rather lengthy question, let's find the intersection points.
1st and 2nd line:
3x+2y=1
3x+2(x-2) = 1
5x = 5
x=1 , then y = 1-2 = -1 ----> point A(1,-1)
2nd and 3rd:
4x - 9y = -22
4x - 9(x-2) = -22
-5x = -40
x = 8 , then y = 2-8 = -6 ----> point B(5,-6)
1st and 3rd:
#1 times 4 --> 12 + 8y = 4
#3 times 3 --> -12x + 27y = 66
add them
35y = 70
y = 2, then
3x + 4 = 1
x = -1 --------> point C(-1,2)
AB = √((5-1)^2 + (-6+1)^2) = √41
AC = √(-1-1)^2 + (2+1)^2) = √13
BC = √((5+1)^2 + (2+6)^2) = √100 = 10
so what is your verdict?
(check my arithmetic)
Well, you copied 5 and 8 wrong from the 2nd and 3rd problem to the final calculation.
Ah, the triangle at right, quite the shape-shifter, isn't it? Well, to determine if it's isosceles, we need to check if any two sides are equal in length. And how do we do that, you ask? With some mathematical hocus pocus, of course!
We start by finding the lengths of the three sides of the triangle. Once we have those, if any two sides are found to be equal, well hot-diggity-dog, we've got ourselves an isosceles triangle!
But hold your horses! Instead of boringly plugging numbers into equations, let's try something a tad more exciting, shall we? How about we turn this into a game of "spot the identicals"?
Now, look closely at those equations, my friend. Do you notice any similarities between the sides? Any sneaky repeating patterns? If you do, jump up and shout "Eureka!" (just make sure your neighbors don't think you're losing your marbles).
If, after some careful observation, you see that the length of one side is equal to the length of another, then congratulations! You've won the game, and the triangle is indeed isosceles.
But if no two sides match up, well, don't despair! You can at least take solace in the fact that you gave it your all and had a giggle along the way.
To determine if the triangle is isosceles, we can start by finding the points of intersection between the three lines.
1. Find the intersection of the lines 3x + 2y = 1 and y = x - 2:
Set the equations equal to each other:
3x + 2y = 1
x - 2 = y
Substitute y = x - 2 into the first equation:
3x + 2(x - 2) = 1
3x + 2x - 4 = 1
5x - 4 = 1
5x = 5
x = 1
Substitute x = 1 into y = x - 2:
y = 1 - 2
y = -1
Therefore, the first point of intersection is (1, -1).
2. Find the intersection of the lines 3x + 2y = 1 and -4x + 9y = 22:
Solve the system of equations by either substitution or elimination.
Let's use substitution:
Solve the second equation for x:
-4x + 9y = 22
x = (9y - 22) / -4
Substitute this into the first equation:
3((9y - 22) / -4) + 2y = 1
(27y - 66) / -4 + 2y = 1
(27y - 66 + 8y) / -4 = 1
(35y - 66) / -4 = 1
Multiplying both sides by -4 to clear the fraction:
35y - 66 = -4
35y = -4 + 66
35y = 62
y = 62 / 35
y = 1.77 (rounded to two decimal places)
Substitute y = 1.77 into x = (9y - 22) / -4:
x = (9(1.77) - 22) / -4
x = (15.93 - 22) / -4
x = -6.07 / -4
x = 1.52 (rounded to two decimal places)
Therefore, the second point of intersection is (1.52, 1.77).
3. Find the intersection of the lines y = x - 2 and -4x + 9y = 22:
Substitute y = x - 2 into the second equation:
-4x + 9(x - 2) = 22
-4x + 9x - 18 = 22
5x - 18 = 22
5x = 40
x = 8
Substitute x = 8 into y = x - 2:
y = 8 - 2
y = 6
Therefore, the third point of intersection is (8, 6).
Now that we have the three points of intersection: (1, -1), (1.52, 1.77), and (8, 6), we can calculate the distances between these points to determine if the triangle is isosceles.
Distance between (1, -1) and (1.52, 1.77):
d1 = sqrt((1.52 - 1)^2 + (1.77 - (-1))^2)
Distance between (1.52, 1.77) and (8, 6):
d2 = sqrt((8 - 1.52)^2 + (6 - 1.77)^2)
Distance between (8, 6) and (1, -1):
d3 = sqrt((8 - 1)^2 + (6 - (-1))^2)
If d1 = d2 or d2 = d3 or d3 = d1, then the triangle is isosceles. If none of the distances are equal, then the triangle is not isosceles.
To determine if the triangle formed by the given equations is isosceles, we need to compare the lengths of its sides. We can find the lengths of the sides by calculating the distances between the points where each pair of lines intersect.
Step 1: Find the intersection points of the lines.
To find the intersection points, we need to solve each pair of equations simultaneously.
For equations 1 and 2:
3x + 2y = 1 ... (Equation 1)
y = x - 2 ... (Equation 2)
Substituting Equation 2 into Equation 1:
3x + 2(x - 2) = 1
3x + 2x - 4 = 1
5x = 5
x = 1
Substituting the value of x into Equation 2:
y = 1 - 2
y = -1
So the intersection point of equations 1 and 2 is (1, -1).
For equations 1 and 3:
3x + 2y = 1 ... (Equation 1)
-4x + 9y = 22 ... (Equation 3)
Solving these equations simultaneously, we get:
3(-4x + 9y) + 2y = 1
-12x + 27y + 2y = 1
-12x + 29y = 1
Substituting Equation 1 into Equation 3:
-12(1) + 29y = 1
-12 + 29y = 1
29y = 13
y = 13/29
Substituting the value of y into Equation 1:
3x + 2(13/29) = 1
3x + 26/29 = 1
3x = 1 - 26/29
3x = (29 - 26)/29
3x = 3/29
x = 1/29
So the intersection point of equations 1 and 3 is (1/29, 13/29).
For equations 2 and 3:
y = x - 2 ... (Equation 2)
-4x + 9y = 22 ... (Equation 3)
Substituting Equation 2 into Equation 3:
-4x + 9(x - 2) = 22
-4x + 9x - 18 = 22
5x = 40
x = 8
Substituting the value of x into Equation 2:
y = 8 - 2
y = 6
So the intersection point of equations 2 and 3 is (8, 6).
Step 2: Calculate the lengths of each side.
Now that we have the coordinates of the three intersection points, we can calculate the lengths of each side of the triangle.
To calculate the distance between two points (x1, y1) and (x2, y2), we use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's calculate the lengths of all three sides:
Side 1:
Length = √((1 - 1)^2 + (-1 - 6)^2) = √(0 + 49) = 7
Side 2:
Length = √((1 - 8)^2 + (-1 - 6)^2) = √((-7)^2 + (-7)^2) = √(49 + 49) = √98 = 7√2
Side 3:
Length = √((8 - 1)^2 + (6 - (-1))^2) = √((7)^2 + (7)^2) = √(49 + 49) = √98 = 7√2
Step 3: Compare the lengths of the sides.
To determine if the triangle is isosceles, we compare the lengths of the sides.
Side 1: 7
Side 2: 7√2
Side 3: 7√2
Since two sides have the same length (both sides 2 and 3 have a length of 7√2), we can conclude that the triangle is isosceles.
Therefore, the triangle formed by the given equations is indeed isosceles.