Acute Angled Triangle Question

Hi All,

In the following question and answer explanation. 1. Can anyone explain why for the Rule 2 part of the explanation below we start at a value of 3, (why not 1 or 2)? 2. How is it known that the values 3,4,5,6,16,17,18,19,20,21 will result in an obtuse angled triangle? I.E. How do we know these values will produce an angle greater than 90degrees?

Thanks in advance,

Question....

If 10, 12 and 'x' are sides of an acute angled triangle, how many integer values of 'x' are possible?

- 7
- 12
- 9
- 13
- 11

Explanatory Answer

Finding the answer to this question requires one to know two rules in geometry.

**Rule 1:** For an acute angled triangle, the square of the LONGEST side MUST BE LESS than the sum of squares of the other two sides.

**Rule 2:** For any triangle, sum of any two sides must be greater than the third side.

The sides are 10, 12 and 'x'.

From Rule 2, x can take the following values: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 – A total of 19 values.

When x = 3 or x = 4 or x = 5 or x = 6, the triangle is an OBTUSE angled triangle (Rule 1 is NOT satisfied).

The smallest value of x that satisfies BOTH conditions is 7. (102 + 72 > 122).

The highest value of x that satisfies BOTH conditions is 15. (102 + 122 > 152).

When x = 16 or x = 17 or x = 18 or x = 19 or x = 20 or x = 21, the triangle is an OBTUSE angled triangle (Rule 1 is NOT satisfied).

Hence, the values of x that satisfy both the rules are x = 7, 8, 9, 10, 11, 12, 13, 14, 15. A total of 9 values.