Question:
Consider obtuse-angled triangles with sides \(8\) cm, \(15\) cm and \(x\) cm. If \(x\) is an integer then how many such triangles exist?
Consider obtuse-angled triangles with sides \(8\) cm, \(15\) cm and \(x\) cm. If \(x\) is an integer then how many such triangles exist?
Show Hint
First use the triangle inequality to find the possible values of \(x\). Then split into two cases: \(15\) is the longest side, or \(x\) is the longest side, and apply the obtuse condition to the longest side.
Updated On: Nov 4, 2026
- 5
- 10
- 21
- 15
- 14
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Triangle inequality:
\(15 - 8 < x < 15 + 8\), so \(7 < x < 23\) and \(x\) can be \(8, 9, \ldots, 22\).
Step 2: Obtuse triangle condition:
A triangle is obtuse when the square of its longest side is greater than the sum of the squares of the other two sides.
Step 3: Case 1, the longest side is 15 (that is, \(8 \le x \le 15\)):
We need \(15^2 > 8^2 + x^2\), that is \(x^2 < 225 - 64 = 161\). Since \(12^2 = 144 < 161\) and \(13^2 = 169 > 161\), we get \(x = 8, 9, 10, 11, 12\). That is \(5\) triangles.
Step 4: Case 2, the longest side is x (that is, \(16 \le x \le 22\)):
We need \(x^2 > 8^2 + 15^2 = 289\). Since \(17^2 = 289\) gives a right triangle (not obtuse), we need \(x \ge 18\): \(x = 18, 19, 20, 21, 22\). That is \(5\) triangles.
Step 5: Add:
\(5 + 5 = 10\).
Final Answer:
The correct option is (2): 10.
\(15 - 8 < x < 15 + 8\), so \(7 < x < 23\) and \(x\) can be \(8, 9, \ldots, 22\).
Step 2: Obtuse triangle condition:
A triangle is obtuse when the square of its longest side is greater than the sum of the squares of the other two sides.
Step 3: Case 1, the longest side is 15 (that is, \(8 \le x \le 15\)):
We need \(15^2 > 8^2 + x^2\), that is \(x^2 < 225 - 64 = 161\). Since \(12^2 = 144 < 161\) and \(13^2 = 169 > 161\), we get \(x = 8, 9, 10, 11, 12\). That is \(5\) triangles.
Step 4: Case 2, the longest side is x (that is, \(16 \le x \le 22\)):
We need \(x^2 > 8^2 + 15^2 = 289\). Since \(17^2 = 289\) gives a right triangle (not obtuse), we need \(x \ge 18\): \(x = 18, 19, 20, 21, 22\). That is \(5\) triangles.
Step 5: Add:
\(5 + 5 = 10\).
Final Answer:
The correct option is (2): 10.
Was this answer helpful?
0
0
Top CAT Quantitative Aptitude Questions
- In a triangle ABC,AB=AC=8cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through B and C. Then the area,in sq.cm,of the overlapping region between the two circles is
- The simple interest accrued on an amount of Rs.25,000 at the end of three years is Rs.7,500. What would be the compound interest accrued on the same amount at the same rate in the same period?
- What is the sum of all roots of the equation \(|x+4|^2–10|x+4|=24\)?
- A solid metal sphere is melted and smaller spheres of equal radii are formed. 10% of the volume of the sphere is lost in the process. The smaller spheres have a radius which is \(\frac{1}{9}\text{th}\) the larger sphere. If 10 litres of paint were needed to paint the larger sphere,how many litres are needed to paint all the smaller spheres?
- The ratio of the ages of a husband and his wife when they got married was 6:5. 4 years and 6 years after their marriage they had their 1st and 2nd children. The sum of the present ages of the husband and wife is 6.4 times the sum of the present ages of their children. The average age of the family at present is 18.5 years. Find the ratio of the ages of the husband and wife when their second child was born.
View More Questions
Top CAT Geometry Questions
In the figure, \(O\) is the centre of the circle and \(AC\) is the diameter. The line \(FEG\) is tangent to the circle at \(E\). If \(\angle GEC = 52^\circ\), find the value of \(\angle E + \angle C\).
- Rekha drew a circle of radius 2 cm on a graph paper of grid 1 cm × 1 cm. She then calculated the area of the circle by adding up only the number of full unit-squares that fell within the perimeter of the circle. If the value that Rekha obtained was 4 sq cm less than the correct value, then find the minimum possible value of \(d\).
- Rekha drew a circle of radius 2 cm on a graph paper of grid 1 cm × 1 cm. She then calculated the area of the circle by adding up only the number of full unit-squares that fell within the perimeter of the circle. If the value that Rekha obtained was 4 sq cm less than the correct value, then find the minimum possible value of d.
what is the minimum possible value of \(d\)? - A rectangle is drawn such that none of its sides has length greater than ‘\( a \)’. All lengths less than ‘\( a \)’ are equally likely. The chance that the rectangle has its diagonal greater than ‘\( a \)’ (in terms of %) is:
- A solid sphere of radius 12 inches is melted and cast into a right circular cone whose base diameter is \( \sqrt{2} \) times its slant height. If the radius of the sphere and the cone are the same, how many such cones can be made and how much material is left out?
View More Questions
Top CAT Questions
- Compare and contrast between Portia and Jessica characters in Merchant of Venice.
- In a triangle ABC,AB=AC=8cm. A circle drawn with BC as diameter passes through A. Another circle drawn with center at A passes through B and C. Then the area,in sq.cm,of the overlapping region between the two circles is
- The simple interest accrued on an amount of Rs.25,000 at the end of three years is Rs.7,500. What would be the compound interest accrued on the same amount at the same rate in the same period?
- What is the sum of all roots of the equation \(|x+4|^2–10|x+4|=24\)?
- A solid metal sphere is melted and smaller spheres of equal radii are formed. 10% of the volume of the sphere is lost in the process. The smaller spheres have a radius which is \(\frac{1}{9}\text{th}\) the larger sphere. If 10 litres of paint were needed to paint the larger sphere,how many litres are needed to paint all the smaller spheres?
View More Questions