Triangle Side Length Bounds
Reasoning
The Triangle Inequality 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 sides of the triangle be 4 cm, 10 cm, and $a$ cm.
Solution Steps
- Apply the theorem to find the upper bound for $a$: The sum of the two known sides must be greater than the third side ($a$).
$4 + 10 > a$
$14 > a$
- Apply the theorem to find the lower bound for $a$: The sum of one known side and the unknown side ($a$) must be greater than the other known side.
$4 + a > 10$
$a > 10 - 4$
$a > 6$
- Combine the inequalities: From steps 1 and 2, we have $a < 14$ and $a > 6$. Combining these gives the possible range for the third side $a$.
$6 < a < 14$
Therefore, the length of the third side $a$ must be strictly between 6 cm and 14 cm.
Similar Questions
- If the sum of the interior angles of a regular polygon is equal to four times the sum of its exterior angles, then what is the number of diagonals in the polygon?
- Find the ratio of the measure of an angle of a regular pentagon to that of a regular octagon.
- The area of an isosceles right angle triangle is $81 \text{ cm}^2$ . Find the length of its hypotenuse.
- In a polygon, the sum of the interior angles is triple the sum of the exterior angles. The number of sides is:
- What is the measure of each exterior angle of a regular octagon?
- The difference between the interior and exterior angles at a vertex of a regular polygon is $140^\circ$. The number of sides of the polygon is:
- The ratio of an interior angle to the exterior angle of a regular polygon is $4:1$. The number of sides of the polygon is:
- If the interior angles of a pentagon are in the ratio 1 : 3 : 5 : 7 : 11, then the measure of the smallest interior angle is:
- If the side of an equilateral triangle is 2 cm, then find the area and the altitude of the triangle.
- If the difference between the interior and exterior angles of a polygon is $36^\circ$, then find the number of sides in the polygon.
Important Questions from Geometry
If the angles of a triangle are in the ratio of 2:3:4, then the difference of the measure of greatest angle and smallest angle is
In ΔABC, ∠A = 90°, AD ┴ BC and AD = BD = 2 cm. The length of CD is
How many lines of symmetry does a rectangle have?
The number of diagonals in each face a cube is
Which of the following is/are the geometric figures with the line of symmetry?
I. Rectangle
II. Isosceles triangle
