The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is:
Question:
The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is:
The lengths of all four sides of a quadrilateral are integer valued. If three of its sides are of length 1 cm, 2 cm and 4 cm, then the total number of possible lengths of the fourth side is:
Options
Answer: 5
Explanation:
Step 1: Recall the quadrilateral inequality For any quadrilateral with sides a, b, c, d, a necessary condition for the quadrilateral to exist is: Sum of any three sides > the fourth side Let the sides be 1, 2, 4, x, where x is the fourth side (integer). Step 2: Apply the inequalities 1 + 2 + 4 > x → 7 > x → x ≤ 6 1 + 2 + x > 4 → 3 + x > 4 → x > 1 1 + 4 + x > 2 → 5 + x > 2 → always true 2 + 4 + x > 1 → 6 + x > 1 → always true Step 3: Combine conditions From the inequalities: x > 1 and x ≤ 6 Possible integer values of x: 2, 3, 4, 5, 6 Total possible values = 5 Answer: 5
Explanation:
Step 1: Recall the quadrilateral inequality For any quadrilateral with sides a, b, c, d, a necessary condition for the quadrilateral to exist is: Sum of any three sides > the fourth side Let the sides be 1, 2, 4, x, where x is the fourth side (integer). Step 2: Apply the inequalities 1 + 2 + 4 > x → 7 > x → x ≤ 6 1 + 2 + x > 4 → 3 + x > 4 → x > 1 1 + 4 + x > 2 → 5 + x > 2 → always true 2 + 4 + x > 1 → 6 + x > 1 → always true Step 3: Combine conditions From the inequalities: x > 1 and x ≤ 6 Possible integer values of x: 2, 3, 4, 5, 6 Total possible values = 5 Answer: 5
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