Question

use $\triangle xyz$, as needed, to complete the given parts.

part 1: let $h$ be the height of $\triangle xyz$.
find the length of the corresponding base.
$\square$ m

part 2: the measure of angle $y$ is $43.69^{\circ}$.
fill in the blank to complete the sine ratio.
$\sin(43.69^{\circ}) = \frac{h}{\square}$

part 3: fill in the blanks in the expression for the area of $\triangle xyz$.
area of $\triangle xyz = \frac{1}{2}\cdot\square\cdot\square\cdot\sin(^{\circ})$ $\text{m}^2$

part 4: write the area of $\triangle abc$ in terms of $a$, $c$, and $b$.
area of $\triangle abc = \square$

Explanation:

Step1: Identify base for height h

The height \( h \) is perpendicular to side \( XY \), so the corresponding base is \( XY \), which has length 13 m.

Step2: Identify hypotenuse for sine ratio

For \( \angle Y \), the height \( h \) is opposite the angle, and the hypotenuse of the right triangle formed is \( YZ \) (length 9 m). So \( \sin(43.69^\circ) = \frac{h}{9} \).

Step3: Apply SAS area formula

The area formula using two sides and included angle is \( \frac{1}{2}ab\sin(C) \). For \( \triangle XYZ \), use sides \( XY=13 \), \( YZ=10 \), and included \( \angle Y=43.69^\circ \). So the expression is \( \frac{1}{2} \cdot 13 \cdot 10 \cdot \sin(43.69^\circ) \).

Step4: Generalize SAS area formula

For \( \triangle ABC \), sides adjacent to \( \angle B \) are \( a \) and \( c \). The area is \( \frac{1}{2}ac\sin(B) \).

Answer:

1. 13
2. 9
3. 13, 10, 43.69
4. $\frac{1}{2}ac\sin(B)$