Question

1. determine the area of this right triangle to the nearest square metre.

Explanation:

Step1: Find side - MN

We know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. In right - triangle $LMN$, $\theta = 57^{\circ}$ and the hypotenuse $LN = 850$ m. Let $MN$ be the side opposite to the angle $57^{\circ}$. So, $\sin57^{\circ}=\frac{MN}{LN}$. Then $MN = LN\times\sin57^{\circ}$. Since $\sin57^{\circ}\approx0.8387$ and $LN = 850$ m, $MN=850\times0.8387 = 712.895$ m.

Step2: Find side - LM

We know that $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. In right - triangle $LMN$, $\cos57^{\circ}=\frac{LM}{LN}$. Then $LM = LN\times\cos57^{\circ}$. Since $\cos57^{\circ}\approx0.5446$ and $LN = 850$ m, $LM = 850\times0.5446=462.91$ m.

Step3: Calculate the area of the right - triangle

The area of a right - triangle is given by $A=\frac{1}{2}\times\text{base}\times\text{height}$. Here, the base can be $LM$ and the height can be $MN$. So, $A=\frac{1}{2}\times LM\times MN$. Substituting the values of $LM$ and $MN$ we found above: $A=\frac{1}{2}\times462.91\times712.895\approx164977$ m².

Answer:

$164977$ m²