Question

I am a little confused: One method you can use to determine whether a triangle is a right triangle, given three side lengths, is to apply the converse of the Pythagorean Theorem. Alternately, you can use trigonometric ratios. Show that the triangle in the diagram is a right triangle by using trigonometric ratios. ( Be sure to show all work/and or reasoning.

Answer 1

Trigonometric ratios are sine, cosine, and tangent (opposite side over hypotenuse, adjacent side over hypotenuse, and opposite side over adjacent side, respectively); if you wanted to prove that one of the angles of the triangle is 90º, then the cosine of that angle would be 0, the sine would be 1, and the tangent would be undefined. 

Answer 2

Answer with explanation:

The triangle in the Diagram Described has following measurement:

 Longest Side = 65 units

One side which can be either Perpendicular or base = 63 units

And , other side which can be also, either Perpendicular or base = 16 units

We can prove that the triangle described is right triangle by two ways.

1. Using Converse of Pythagorean Theorem

Square of Longest side = Sum of Squares of other two sides-----(1)

So, Square of Longest Side =  65²=4225

Sum of Square of other two sides = 16² + 63²

                                        = 256 + 3969

                                        =  4225

Statement (1), is valid.

So,Triangle is right angled triangle, right angled at A.

2. using Trigonometric Ratios

Suppose the triangle is right Angled at A.

In Right triangle B AC

$tan B=\frac{\text{Perpendicular}}{\text{Base}}\\\\tan B=\frac{16}{63}\\\\tan C=\frac{63}{16}\\\\ tan(B +C)=\frac{tan B + tan C}{1-tan B \times tanC}\\\\tan (B +C)=\frac{\frac{16}{63}+\frac{63}{16}}{1-\frac{16}{63}\times \frac{63}{16}}\\\\tan (B +C)=\frac{\text{Any rational number}}{0}\\\\tan (B +C)=\infty\\\\B +C=90^{\circ}\\\\ \text{Using the trigonometric Identity},tan(A+B)=\frac{tan A +tan B}{1-tan A*tan B}$

B +C =90°

Also,→ ∠A + ∠B + ∠C=180°≡ (Angle sum property of triangle)

→∠A +90°=180°

→∠A=180° -90°

→∠A=90°

So, triangle is right Angled triangle , Right angled at A.

Hence ,proved.