Question

which triangle is an isosceles triangle?

• triangle with sides 7, 9, 5
• right triangle with sides 15, 12, 9
• triangle with sides 3, 3, 3
• triangle with sides 6, 6, 4

Explanation:

Step1: Recall isosceles triangle definition

An isosceles triangle has at least two sides of equal length.

Step2: Analyze first triangle

Sides: 7, 5, 9. All side lengths are distinct (\(7
eq5\), \(7
eq9\), \(5
eq9\)), so it is a scalene triangle.

Step3: Analyze second triangle

Sides: 12, 9, 15. Check \(12^2 + 9^2=144 + 81 = 225=15^2\), so it is a right triangle. But \(12
eq9\), \(12
eq15\), \(9
eq15\), so it is a scalene right triangle.

Step4: Analyze third triangle

Sides: 3, 3, 3. All three sides are equal. By definition, an equilateral triangle is a special case of an isosceles triangle (since it has at least two equal sides, in fact three). So this is an isosceles (and equilateral) triangle.

Step5: Analyze fourth triangle

Sides: 6, 6, 4. Two sides have length 6, so it has at least two equal sides. Thus, it is an isosceles triangle.

Answer:

The fourth triangle (with side lengths 6, 6, 4) and the third triangle (with side lengths 3, 3, 3) are isosceles triangles (the third is also equilateral, a special case of isosceles). But if we consider the options, the fourth one (and third) - let's analyze each:

1. First triangle: sides 7, 5, 9 - all different, scalene.
2. Second triangle: sides 12, 9, 15 (check \(12^2 + 9^2 = 144 + 81 = 225 = 15^2\), right triangle, sides 12,9,15 - all different, scalene.
3. Third triangle: sides 3,3,3 - equilateral (a type of isosceles with all sides equal).
4. Fourth triangle: sides 6,6,4 - two sides equal, isosceles.

If we have to pick the non - equilateral isosceles (or including equilateral), the third (3,3,3) and fourth (6,6,4) are isosceles. But likely the intended answer is the fourth (or third). Let's confirm the definition: an isosceles triangle has at least two equal sides.

So the triangles with side lengths 3,3,3 (third) and 6,6,4 (fourth) are isosceles. If we take the fourth one (the last option with sides 6, 6, 4) or the third (3,3,3).