The number of triangles with integer sides and with perimeter 15 is ………… .
The number of triangles with integer sides and with perimeter 15 is ………… .
Show Hint
Solution and Explanation
Understanding the Triangle Inequality: For a triangle with sides \(a\), \(b\), and \(c\), the triangle inequality must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Finding the Combinations:
- Start with the largest possible side: Let's assume the largest side is 7. Then the remaining two sides must sum to 8. Possible combinations are (7, 7, 1), (7, 6, 2), (7, 5, 3), (7, 4, 4). All of these satisfy the triangle inequality.
- Next largest side is 6: Remaining two sides sum to 9. Possible combinations are (6, 6, 3), (6, 5, 4).
- Next largest side is 5: Remaining two sides sum to 10. The possible combination is (5, 5, 5).
Checking for Duplicates: We have listed all possible combinations without repeating any.
Counting the Triangles: We have found 7 unique combinations:
- (7, 7, 1)
- (7, 6, 2)
- (7, 5, 3)
- (7, 4, 4)
- (6, 6, 3)
- (6, 5, 4)
- (5, 5, 5)
Answer: The number of triangles with integer sides and perimeter 15 is 7.
Top Questions on Geometry
- The equation of the circle with extremities \( (1, 3) \) and \( (5, 7) \) of the diameter is:
- Find the area of the triangle formed by the lines: $ y = -4, \quad y = x, \quad y = -4 $
- Let $ A(4, -2), B(1, 1) $ and $ C(9, -3) $ be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is __________.
- Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles \( ABC, ACD, \) and \( ADB \) be 5, 6 and 7 square units respectively. Then the area (in square units) of the tetrahedron ABCD is equal to:
- Find the equation of the circle touching the x-axis at \( (9, 0) \) and the line \( y = 14 \).
Questions Asked in IPMAT exam
The sentences given below, when properly sequenced, form a coherent paragraph. Each sentence is labelled with a number. Decide on the most logical order and enter the sequence of numbers in the space provided. Only numbers are to be entered in the space provided for the answer, and no letters, characters, or spaces should be entered. For example, a response such as 3412 is valid, and responses such as 3412. or 3 41 2 are invalid.
1. In this microgravity environment, your blood also tends to move towards the heart and head.
2. Both effects are only short-lived and, after a brief period of readjustment on arriving back home, the heart and spine return to normal.
3. The heart interprets this as an increase in the amount of blood in the body and that it needs to pump less, so it shrinks.
4. In space with less gravitational force than on Earth, there's less pressure on your spine and so it'll get a bit longer, effectively making you as much as two inches taller.- IPMAT - 2024
- Paragraph
The number of factors of 1800 that are multiple of 6 is …………. .
- IPMAT - 2024
- Simplification
The number of real solutions of the equation \((x^2 - 15x + 55)^{x^2 - 5x + 6} = 1\) is __________.
- IPMAT - 2024
- Algebra
- The highest possible age of an employee of company A is .......................
- IPMAT - 2024
- Statistics
In a group of 150 students, 52 like tea, 48 like juice, and 62 like coffee. If each student in the group likes at least one among tea, juice, and coffee, then the maximum number of students that like more than one drink is _______________.
- IPMAT - 2024
- Venn Diagrams