If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7,

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Question: If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the perimeter of the triangle?

Options:

Correct Answer: 14

Solution:

Perimeter = AB + BC + CA = √((4-1)² + (5-1)²) + √((7-4)² + (2-5)²) + √((1-7)² + (1-2)²) = 3 + √(9 + 9) + √(36 + 1) = 3 + 4.24 + 6.08 = 13.32.

If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the perimeter of the triangle?

If the coordinates of the vertices of a triangle are A(1, 1), B(4, 5), and C(7, 2), what is the perimeter of the triangle?

Step 1: Identify the coordinates of the triangle's vertices. A(1, 1), B(4, 5), C(7, 2).
Step 2: Calculate the length of side AB using the distance formula: AB = √((x2 - x1)² + (y2 - y1)²). Here, A(1, 1) and B(4, 5).
Step 3: Substitute the coordinates into the formula: AB = √((4 - 1)² + (5 - 1)²).
Step 4: Simplify the calculation: AB = √(3² + 4²) = √(9 + 16) = √25 = 5.
Step 5: Calculate the length of side BC using the same distance formula: BC = √((7 - 4)² + (2 - 5)²).
Step 6: Substitute the coordinates into the formula: BC = √((7 - 4)² + (2 - 5)²).
Step 7: Simplify the calculation: BC = √(3² + (-3)²) = √(9 + 9) = √18 = 4.24.
Step 8: Calculate the length of side CA using the distance formula: CA = √((1 - 7)² + (1 - 2)²).
Step 9: Substitute the coordinates into the formula: CA = √((-6)² + (-1)²).
Step 10: Simplify the calculation: CA = √(36 + 1) = √37 = 6.08.
Step 11: Add the lengths of all sides to find the perimeter: Perimeter = AB + BC + CA.
Step 12: Substitute the lengths: Perimeter = 5 + 4.24 + 6.08.
Step 13: Calculate the total: Perimeter = 15.32.

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.