If the coordinates of the vertices of a triangle are (1, 2), (4, 6), and (7, 2),

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Question: If the coordinates of the vertices of a triangle are (1, 2), (4, 6), and (7, 2), what is the perimeter of the triangle?

Options:

Correct Answer: 14

Solution:

Perimeter = AB + BC + CA = √[(4-1)² + (6-2)²] + √[(7-4)² + (2-6)²] + √[(1-7)² + (2-2)²] = 3 + √(9 + 16) + 6 = 3 + 5 + 6 = 14.

If the coordinates of the vertices of a triangle are (1, 2), (4, 6), and (7, 2), what is the perimeter of the triangle?

If the coordinates of the vertices of a triangle are (1, 2), (4, 6), and (7, 2), what is the perimeter of the triangle?

Correct Answer: 14

Step 1: Identify the coordinates of the triangle's vertices. They are A(1, 2), B(4, 6), and C(7, 2).
Step 2: Calculate the distance between points A and B using the distance formula: AB = √[(x2 - x1)² + (y2 - y1)²]. Here, (x1, y1) = (1, 2) and (x2, y2) = (4, 6).
Step 3: Substitute the values into the formula: AB = √[(4 - 1)² + (6 - 2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
Step 4: Calculate the distance between points B and C using the same distance formula: BC = √[(7 - 4)² + (2 - 6)²].
Step 5: Substitute the values: BC = √[(7 - 4)² + (2 - 6)²] = √[3² + (-4)²] = √[9 + 16] = √25 = 5.
Step 6: Calculate the distance between points C and A: CA = √[(1 - 7)² + (2 - 2)²].
Step 7: Substitute the values: CA = √[(-6)² + 0²] = √[36 + 0] = √36 = 6.
Step 8: Add the lengths of all sides to find the perimeter: Perimeter = AB + BC + CA = 5 + 5 + 6.
Step 9: Calculate the total: 5 + 5 + 6 = 16.

Distance Formula – The question tests the understanding of the distance formula to calculate the lengths of the sides of a triangle given its vertices.
Perimeter Calculation – It assesses the ability to sum the lengths of the sides to find the perimeter of the triangle.