In ABC, if a=36 cm and c=23degrees, then what is the length of altitude h? Round your answer to the nearest whole number

Answer 1

To find the altitude $h$ of triangle ABC, where $a=36$ cm and $\mathrm{\angle }C={23}^{\circ }$, we can use the formula for the area of a triangle, which can also be expressed in terms of the base and height:

$$\text{Area}=\frac{1}{2}\times a\times h$$

We can also use the formula for the area in terms of two sides and the included angle:

$$\text{Area}=\frac{1}{2}\times b\times c\times \sin (A)$$

In this case, since we only have one side (a) and the angle C, we can modify our method. The side $a$ will serve as the base (let's say $b$).

To find the altitude $h$ for side $a$, we can use the following formula relating the altitude and side $a$:

$$h=a\cdot \sin (C)$$

1. First, we need to compute $\sin ({23}^{\circ })$:

Using a calculator, $$\sin ({23}^{\circ })\approx 0.3907$$

2. Now plug this value into the formula for $h$:

$$h=36,\text{cm}\cdot \sin ({23}^{\circ })\approx 36,\text{cm}\cdot 0.3907\approx 14.0852,\text{cm}$$

3. Finally, rounding $h$ to the nearest whole number:

$$h\approx 14,\text{cm}$$

Thus, the length of the altitude $h$ is approximately 14 cm.