Which statements about the diagram are true? Select three options. DE + EF > DF \triangle DEF is an isosceles triangle. 5 < DF < 13 DE + DF < EF \triangle DEF is a scalene triangle
The Correct Answer and Explanation is:
Based on the triangle diagram shown and the question, we are given:
- DE=4DE = 4DE=4 units
- EF=9EF = 9EF=9 units
- DFDFDF is unknown, but it is marked as congruent to DEDEDE, meaning DF=4DF = 4DF=4 units
- We are to choose three true statements.
Step-by-step Analysis:
- Statement: DE + EF > DF Plug in the values:
DE+EF=4+9=13DE + EF = 4 + 9 = 13DE+EF=4+9=13
DF=4DF = 4DF=4
Since 13>413 > 413>4, this is true. - Statement: △DEF is an isosceles triangle. DE=DF=4DE = DF = 4DE=DF=4, and EF=9EF = 9EF=9. Since two sides are equal, triangle DEF is isosceles.
This is true. - Statement: 5 < DF < 13 We already know DF=4DF = 4DF=4, so this statement becomes:
5<4<135 < 4 < 135<4<13, which is false. - Statement: DE + DF < EF DE+DF=4+4=8DE + DF = 4 + 4 = 8DE+DF=4+4=8, and EF=9EF = 9EF=9
8<98 < 98<9, so this is true. - Statement: △DEF is a scalene triangle. A scalene triangle has all sides of different lengths. Since DE=DF=4DE = DF = 4DE=DF=4, it is not scalene.
This is false.
✅ Correct Answers:
- DE + EF > DF
- △DEF is an isosceles triangle
- DE + DF < EF
Explanation
The triangle shown in the diagram is labeled △DEF\triangle DEF△DEF, with side lengths DE=4DE = 4DE=4, EF=9EF = 9EF=9, and DF=4DF = 4DF=4. Because two of its sides are equal in length (DE=DFDE = DFDE=DF), the triangle is classified as isosceles, not scalene. A scalene triangle requires all three sides to be of different lengths, which is not the case here.
The inequality DE+EF>DFDE + EF > DFDE+EF>DF holds because 4+9=134 + 9 = 134+9=13, and 13>413 > 413>4. This aligns with the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be greater than the third side. Similarly, the inequality DE+DF<EFDE + DF < EFDE+DF<EF evaluates to 4+4=84 + 4 = 84+4=8, and since 8<98 < 98<9, this is also a valid and true statement.
On the other hand, the statement 5<DF<135 < DF < 135<DF<13 is incorrect because DF=4DF = 4DF=4, which does not satisfy 5<DF5 < DF5<DF. Therefore, this option is false. Additionally, since two sides of the triangle are equal, the triangle cannot be scalene, making the final statement false as well.
In summary, the three correct and true statements based on the diagram and triangle properties are:
- DE+EF>DFDE + EF > DFDE+EF>DF
- △DEF\triangle DEF△DEF is an isosceles triangle.
- DE+DF<EFDE + DF < EFDE+DF<EF
These conclusions are grounded in both direct measurements and geometric principles.
