Question

Which statements are true regarding triangle LMN? Check all that apply.

Answer 1

Answer:

1, 3, 5

Step-by-step explanation:

Answer 2

Answer:

$NM = x$

$LM = x\sqrt{2}$

$tan (45) = 1$

Step-by-step explanation:

Step 1: Pythagoras Theorem

Pythagoras theorem relates the three sides of the triangle in such a way that the sum of the square of base and perpendicular is equal to hypotenuse, such as:

                                        $LM^{2} =LN^{2} +NM^{2}$

Step 2: Trigonometric Functions

Only for a right angle triangle following three trigonometric relations are valid

                                        $sin (\theta) = \frac{opposite}{hypotenuse}$

                                        $cos (\theta) = \frac{adjacent}{hypotenuse}$

                                    $tan (\theta)=\frac{sin (\theta)}{cos (\theta)} = \frac{opposite}{adjacent}$

Step 3: Verifying all the possible answers

A: Since, LN = x and using $tan (45) =1$

we can calculate

                                              $tan (\theta)= \frac{opposite}{adjacent}$

                                           $tan (45)= \frac{NM}{x} =1$

therefore, NM = x (true)

B: As NM = x therefore it can not be equal to $x\sqrt{2\\}$.

C: Using Pythagoras Theorem

                                        $LM^{2} =LN^{2} +NM^{2}$

                                           $LM^{2} =x^{2} +x^{2}$

                                              $LM^{2} =2x^{2}$

                                         $LM = \sqrt{2x^{2}} = x\sqrt{2}$

It can also be proved using trigonometric relation

                                           $cos (45) = \frac{x}{LM}$

                                            $LM = \frac{x}{cos (45)}$

As, $\frac{1}{cos (45)}= \sqrt{2}$

Therefore

                                            $LM = x\sqrt{2}$

D and E:

Using same approach similar to part A

Since, LN = x and NM = x

we can calculate

                                              $tan (\theta)= \frac{opposite}{adjacent}$

                                           $tan (45)= \frac{x}{x} =1$

Therefore, $tan (45) = 1$  and not equal to $\frac{\sqrt{2} }{2}$