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Okay, so I need to figure out how to solve this problem where I have to find the area of a triangle with vertices at (1,2), (3,4), and (5,6). Hmm, let me remember. I think there's a formula for the area of a triangle when you know the coordinates of the vertices. Was it the shoelace formula? Yeah, that sounds right. Let me recall how that works.
Alright, the shoelace formula. If I have three points, let's call them (x1, y1), (x2, y2), and (x3, y3), then the area is half the absolute value of the determinant of a matrix formed by these points. Or maybe it's calculated by multiplying coordinates in a certain way and subtracting. Let me write it down step by step.
First, I'll list the coordinates in order. Let's assign the points: A(1,2), B(3,4), and C(5,6). To apply the shoelace formula, I need to arrange the points either clockwise or counter-clockwise and then repeat the first point at the end to close the polygon. So, writing them out:
x1, y1 = 1, 2
x2, y2 = 3, 4
x3, y3 = 5, 6
Then, repeat the first point:
x4, y4 = 1, 2
Now, the formula is to take the sum of the products of each xi and yi+1 minus the sum of the products of each yi and xi+1, then take half the absolute value. So, mathematically:
Area = (1/2) | (x1y2 + x2y3 + x3y4) - (y1x2 + y2x3 + y3x4) |
Plugging in the numbers:
First part: (14) + (36) + (5*2) = 4 + 18 + 10 = 32
Second part: (23) + (45) + (6*1) = 6 + 20 + 6 = 32
Subtracting the two: 32 - 32 = 0
Taking absolute value and half of it: (1/2)|0| = 0
Wait, that can't be right. A triangle with area zero? That means the points are colinear. But (1,2), (3,4), (5,6) – these points are on the line y = x + 1, right? Let me check.
For point A: x=1, y=2 → 1 +1 = 2 ✔️
Point B: 3 +1 =4 ✔️
Point C:5 +1=6 ✔️
Oh! So all three points lie on the same straight line. That's why the area is zero. So the area of the triangle is zero because the points are colinear, forming a degenerate triangle.
But let me make sure I didn't make a mistake in the calculation. Let me go through the shoelace formula again.
First sum: (14) + (36) + (5*2) = 4 + 18 +10 =32
Second sum: (23) + (45) + (6*1) =6 +20 +6=32
32 -32=0. Absolute value of 0 is 0. Half of 0 is 0. Yep, that's correct.
Alternatively, maybe I can use another method to verify. For example, the area of a triangle can also be calculated using the base times height divided by two. But if the points are colinear, the height would be zero, so the area is zero. That matches.
Alternatively, using vectors. The vectors AB and AC. If they are parallel, the area is zero.
Vector AB is (3-1, 4-2) = (2,2)
Vector AC is (5-1,6-2) = (4,4)
AC is exactly 2 times AB, so they are colinear. Hence, the area is zero.
Another way: the slope between A and B is (4-2)/(3-1)=2/2=1
Slope between B and C is (6-4)/(5-3)=2/2=1
Same slope, so they are colinear.
Therefore, all methods confirm that the area is zero. So the answer is 0.
But just to be thorough, let me try the determinant method. The area is (1/2)| determinant of the matrix formed by the coordinates.
The determinant formula is:
Area = (1/2)| x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |
Plugging in:
1*(4 -6) + 3*(6 -2) +5*(2 -4) =1*(-2) +3*(4) +5*(-2) = -2 +12 -10 =0
Multiply by 1/2 and absolute value: 0. Yep, same result.
So all methods lead to the conclusion that the area is zero because the three points are colinear. Therefore, the triangle is degenerate, and its area is zero.
The area of the triangle with vertices at (1,2), (3,4), and (5,6) is calculated using the shoelace formula. However, since all three points lie on the straight line ( y = x + 1 ), they are collinear, forming a degenerate triangle with no area.
Step-by-Step Explanation:
Verify Collinearity:
Check if the slope between consecutive points is the same.
Slope between (1,2) and (3,4): ( \frac{4-2}{3-1} = 1 )
Slope between (3,4) and (5,6): ( \frac{6-4}{5-3} = 1 )
Equal slopes confirm collinearity.
Shoelace Formula Application:
Coordinates: ( (1,2) ), ( (3,4) ), ( (5,6) ), repeated: ( (1,2) )
Compute sums:
( (1 \cdot 4 + 3 \cdot 6 + 5 \cdot 2) = 4 + 18 + 10 = 32 )
( (2 \cdot 3 + 4 \cdot 5 + 6 \cdot 1) = 6 + 20 + 6 = 32 )
Subtract sums: ( 32 - 32 = 0 )
Area: ( \frac{1}{2} |0| = 0 )
https://www.18122.com/url/usdt/usdt2141.jpg
Alternative Verification (Vectors/Determinant):
Vectors ( \overrightarrow{AB} = (2,2) ) and ( \overrightarrow{AC} = (4,4) ) are scalar multiples.
Determinant method yields ( 0 ).
Final Answer:
The area of the triangle is (\boxed{0}) because the vertices are collinear.
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