Solve triangle X Y Z given ∠X=90^∘, ∠Y=23^∘ 17^' and Y Z=20.0 mm. Determine also its area.It is

step 1 Okay, the question is sold the triangle x, y, z given as we have given the angle x is equal to 9

Solve triangle X Y Z given ∠X=90^∘, ∠Y=23^∘ 17^' and Y...

Solve triangle $X Y Z$ given $\angle X=90^{\circ}, \angle Y=23^{\circ} 17^{\prime}$ and $Y Z=20.0 \mathrm{~mm}$. Determine also its area.It is always advisable to make a reasonably accurate sketch so as to visualise the expected magnitudes of unknown sides and angles. Such a sketch is shown in Fig. 12,16 . $$ \begin{aligned} \angle Z &=180^{\circ}-90^{\circ}-23^{\circ} 17^{\prime}=66^{\circ} 43^{\prime} \\ \sin 23^{\circ} 17^{\prime} &=\frac{X Z}{20.0} \end{aligned} $$

Solve triangle $X Y Z$ given $\angle X=90^{\circ}, \angle Y=23^{\circ} 17^{\prime}$ and $Y Z=20.0 \mathrm{~mm}$. Determine also its area.It is always advisable to make a reasonably accurate sketch so as to visualise the expected magnitudes of unknown sides and angles. Such a sketch is shown in Fig. 12,16 .
$$
\begin{aligned}
\angle Z &=180^{\circ}-90^{\circ}-23^{\circ} 17^{\prime}=66^{\circ} 43^{\prime} \\
\sin 23^{\circ} 17^{\prime} &=\frac{X Z}{20.0}
\end{aligned}
$$

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Key Concepts

Area Calculation of a Triangle

The area of a triangle can be calculated using different methods, including the formula 0.5 × base × height for right triangles, or by employing trigonometric formulas when angles and sides are known. This concept is crucial in connecting geometric dimensions to measurable quantities.

Solving a Right Triangle

In a right triangle, knowing one acute angle and one side provides enough information to determine all remaining sides and angles through the application of trigonometric ratios and the angle sum property. This systematic approach is central to many geometric problems.

Trigonometric Ratios in Right Triangles

Trigonometric functions like sine, cosine, and tangent serve as vital tools in linking angles with side lengths in right triangles. They enable the calculation of unknown sides or angles, given at least one side length and one acute angle.

Angle Sum Property of Triangles

Every triangle's interior angles add up to 180°. This fundamental principle is used to compute an unknown angle when the other two angles are known, ensuring the consistency of triangular geometry.

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