SAT I Quant : Geometry : Scalene Triangles

Question

How many scalene triangles with integer sides having a perimeter of 15 units exist, if one of its sides measures 3 units?
A. 3
B. 5
C. 1
D. 0
E. 7

Correct Answer : Choice C. Only 1 triangle.

Explanatory Answer

Data given
a. The triangle is a scalene triangle. The measure of the 3 sides are different.
b. Perimeter is 15. If the sides measure a, b and c units, then a + b + c =15.
c. One of its sides is 3. Let us say a = 3.
d. Measure of all sides are integers.

Combining points a and b, we get b + c = 12.

Values that b and c can take such that their sum is 12 and b is not equal to c are listed below
1. 1 and 11
2. 2 and 10
3. 3 and 9
4. 4 and 8
5. 5 and 7

There is one more step to go before we count the possibilities.

In any triangle, sum of two of its sides will be greater than the third side.

1. If b and c were 1 and 11, the 3 sides of the triangle will be 1, 3 and 11. 1 + 3 is not greater than 11.
So, b and c CANNOT be 1 and 11.

2. If b and c were 2 and 10, the 3 sides will be 2, 3 and 10. 2 + 3 is not greater than 10. So, (2, 3, 10) IS NOT a possibility.

3. If b and c were 3 and 9, the sides of the triangle are 3, 3 and 9. However, the 3 sides are not different. So, this combination IS NOT possible.

4. If b and c were 4 and 8, the sides of the triangle are 3, 4 and 8. 3 + 4 is not greater than 8. So, (3, 4, 8) is NOT A TRIANGLE.

5. If b and c were 5 and 7, the sides of the triangle are 3, 5 and 7. 3 + 5 > 7. All 3 sides are different. So, (3, 5, 7) is THE ONLY POSSIBLE Triangle.

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