Chapter 8, Problem 3CT

To determine

To complete: the statement based on the given figure.

Answer to Problem 3CT

ΔDNF ~ $\Delta DNE$ ~ $\Delta NEF$

Explanation of Solution

Given Information: A triangle with a right angle and a perpendicular directed to the hypotenuse. The measure of sides is marked.

For a right triangle with a perpendicular being drawn to the hypotenuse (as shown below),the both small triangles are congruent to each other and to the bigger triangle.

  

In the given triangle ΔDNF, with right angle $\angle DNF$ , the perpendicular is drawn from N to hypotenuse DF. This gives the relation between the triangles as,

ΔDNF ~ $\Delta DNE$

  $\Delta DNE$ ~ $\Delta NEF$

ΔDNF ~ $\Delta NEF$

From this, the relation relevant to the present case can be written as,

ΔDNF ~ $\Delta DNE$ ~ $\Delta NEF$