Question

which statements about the diagram are true? select three options. de + ef>df △def is an isosceles triangle. 5<df<13 de + df<ef △def is a scalene triangle.

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle is greater than the length of the third side. For \(\triangle DEF\) with \(DE = 4\) and \(EF=9\), we have \(DE + EF>DF\), \(DE + DF>EF\), and \(EF + DF>DE\).

Step2: Analyze the type of triangle

Since \(DE = 4\) and \(EF = 9\) and the third - side length \(DF\) satisfies the triangle - inequality: \(|DE - EF|eq4\) and \(DF
eq9\)), \(\triangle DEF\) is a scalene triangle.

Step3: Check each option

• Option 1: \(DE + EF>DF\) is true by the triangle - inequality theorem.
• Option 2: \(\triangle DEF\) is not an isosceles triangle as no two side - lengths are equal.
• Option 3: \(5
• Option 4: \(DE+DFEF\).
• Option 5: \(\triangle DEF\) is a scalene triangle as all side - lengths are different.

Answer:

DE + EF>DF, 5<DF<13, \(\triangle DEF\) is a scalene triangle.