Which of the following triplets does not represent the sides of a triangle?
(2, 3, 6)
Checking Triangle Sides using the Triangle Inequality Theorem
To determine if three given lengths can form the sides of a triangle, we use the fundamental principle called the Triangle Inequality Theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.
Let the three side lengths be denoted by \(a\), \(b\), and \(c\). For these lengths to form a triangle, the following three conditions must all be true:
- Condition 1: \(a + b > c\)
- Condition 2: \(a + c > b\)
- Condition 3: \(b + c > a\)
If even one of these conditions is not met, then the three lengths cannot form a triangle. Let's check each given triplet against these conditions.
Analyzing Option 1: (3, 4, 5)
Here, \(a=3\), \(b=4\), \(c=5\).
- \(3 + 4 > 5 \implies 7 > 5\) (True)
- \(3 + 5 > 4 \implies 8 > 4\) (True)
- \(4 + 5 > 3 \implies 9 > 3\) (True)
All conditions are satisfied. This triplet can form a triangle (specifically, a right-angled triangle).
Analyzing Option 2: (4, 7, 10)
Here, \(a=4\), \(b=7\), \(c=10\).
- \(4 + 7 > 10 \implies 11 > 10\) (True)
- \(4 + 10 > 7 \implies 14 > 7\) (True)
- \(7 + 10 > 4 \implies 17 > 4\) (True)
All conditions are satisfied. This triplet can form a triangle.
Analyzing Option 3: (3, 6, 8)
Here, \(a=3\), \(b=6\), \(c=8\).
- \(3 + 6 > 8 \implies 9 > 8\) (True)
- \(3 + 8 > 6 \implies 11 > 6\) (True)
- \(6 + 8 > 3 \implies 14 > 3\) (True)
All conditions are satisfied. This triplet can form a triangle.
Analyzing Option 4: (2, 3, 6)
Here, \(a=2\), \(b=3\), \(c=6\).
- \(2 + 3 > 6 \implies 5 > 6\) (False)
- \(2 + 6 > 3 \implies 8 > 3\) (True)
- \(3 + 6 > 2 \implies 9 > 2\) (True)
The first condition, \(a + b > c\), is not satisfied because \(2 + 3 = 5\), which is not greater than 6. Since one of the conditions is false, this triplet cannot form a triangle.
Conclusion on Triangle Side Triplet
Based on the Triangle Inequality Theorem, the triplet that does not represent the sides of a triangle is (2, 3, 6) because the sum of two sides (2 and 3) is not greater than the third side (6).
| Triplet (a, b, c) | Check 1 (a+b > c) | Check 2 (a+c > b) | Check 3 (b+c > a) | Forms a Triangle? |
|---|---|---|---|---|
| (3, 4, 5) | \(3+4=7 > 5\) (True) | \(3+5=8 > 4\) (True) | \(4+5=9 > 3\) (True) | Yes |
| (4, 7, 10) | \(4+7=11 > 10\) (True) | \(4+10=14 > 7\) (True) | \(7+10=17 > 4\) (True) | Yes |
| (3, 6, 8) | \(3+6=9 > 8\) (True) | \(3+8=11 > 6\) (True) | \(6+8=14 > 3\) (True) | Yes |
| (2, 3, 6) | \(2+3=5 > 6\) (False) | \(2+6=8 > 3\) (True) | \(3+6=9 > 2\) (True) | No |
Additional Information: Triangle Properties
The Triangle Inequality Theorem is a fundamental property in Euclidean geometry. It directly relates the lengths of the sides of a triangle. This theorem has implications for understanding distances in geometry. For example, the shortest distance between two points is a straight line, and any path that deviates (like going via a third point forming a triangle) will be longer.
Types of triangles are classified based on their side lengths (equilateral, isosceles, scalene) or angles (acute, obtuse, right-angled). The triplet (3, 4, 5) is a classic example of the sides of a right-angled triangle, as it satisfies the Pythagorean theorem (\(3^2 + 4^2 = 5^2\)). However, satisfying the Triangle Inequality Theorem is a requirement for any triangle, regardless of its angles or specific type.
Similar Questions
ABCD is a trapezium in which AB is parallel to DC. Let E and F be the midpoints on AD and BC respectively. If EF = 10 cm and AB - DC = 4 cm, then what is the value of AB × DC?
The area of a sector of a circle of radius 4 cm is 25.6 cm 2. What is the radian measure of the arc of the sector?
- X, Y and Z are three equilateral triangles. The sum of the areas of X and Y is equal to the area of Z. If the side lengths of X and Y are 6 cm and 8 cm respectively, then what is the side length of Z?
If the perimeter of a right-angled triangle is 30 cm and the hypotenuse is 13 cm, then what is the area of the triangle?
The area of a rhombus is 336 square cm. If the length of one of its diagonals is 48 cm, then what is the perimeter of the rhombus ?
The lengths of the sides of a right-angled triangle are consecutive even integers (in cm). What is the product of these integers?
The arch of a bridge is in the form of an arc of a circle. If the span of the bridge is 40 m and height in the middle is 8 m, then what is the radius of curvature of the bridge?
ABC is a triangle with sides AB = 41 cm, BC = 28 cm and CA = 15 cm. If D, E and F are the mid-points of AB, BC and CA respectively, then what is the area of the triangle DEF?
What is the area of the circle (approximately) inscribed in a triangle with side lengths 12 cm, 16 cm and 20 cm ?
How long will a man take to walk around the boundary of a square field of area 25 hectares at the rate of 5 km/hr ?
Important Questions from Plane Figures
If the area of a square is 625 cm 2, then what is the perimeter of the square?
The area and the perimeter of a sheet of paper are 240 cm 2and 68 cm, respectively. What would be its length and breadth?
One side of rectangular field is 15 meters and one of its diagonals is 17 meters. Then find the area of the field.
The bisector of ∠B in ΔABC meets AC at D. If AB = 12 cm, BC = 18 cm and AC = 15 cm, then the length of AD (in cm) is:
The perimeter and the length of one of the diagonals of a rhombus is 26 cm and 5 cm respectively. Find the length of its other diagonal (in cm).
