Question

Solving a Right

Triangle Solve the right triangle

Short answer

Base of triangle is 16, the hypotenuse is$\mathbf{16}\sqrt{\mathbf{2}}$ and the remaining angle is${\mathbf{45}}^{\mathbf{\circ }}$.

Step-by-step solution

Step 1. Given information

A right triangle with known value of a side of and an angle of${45}^{\circ }$.

Step 2. Concept used

A side and an angle are both known. Use trigonometric formulas to get the unknown side. And use triangle sum property of angles to find unknown ngle. Use formulae as-

$\begin{array}{l}\Rightarrow \sin \theta =\frac{p}{h}\\ \Rightarrow \cos \theta =\frac{b}{h}\\ \Rightarrow \tan \theta =\frac{p}{b}\end{array}$

Where

$\begin{array}{l}h=hypotenuse\\ b=base\\ p=perpendicular\end{array}$

Step 3. Calculation

Now use the formulae,

$\begin{array}{l}\tan \theta =\frac{b}{a}\\ \Rightarrow \tan 45=\frac{16}{x}\\ \Rightarrow 1=\frac{16}{x}\\ \Rightarrow x=16\end{array}$

Use sum of triangle’s angle,

$\begin{array}{l}\angle A+\angle B+\angle C={180}^{\circ }\\ \angle A+45+90=180\\ \angle A=180-135\\ \angle A={45}^{\circ }\end{array}$

Use Pythagoras theorem,

$\begin{array}{l}{c}^2=a^2+b^2\\ c^2=\left(16\right){}^2+{\left(16\right)}^2\\ c^2=256\\ c=16\sqrt{2}\end{array}$