Question

1. un arpenteur a tracé le croquis suivant à l’aide des mesures qu’il a effectuées. les points ont été localisés par coordonnées rectangulaires.

a. calculer l’aire du terrain triangulaire attenant au ruisseau.
b. calculer l’aire de la parcelle de terrain boisée au bas du croquis.
82.4 m
35.9 m
24.5 m
68.4 m
18.3 m
108.95 m
104.28°

Explanation:

Part (a)

Step1: Identify triangle formula

The triangular terrain by the stream: we can use the formula for the area of a triangle \( A=\frac{1}{2}\times base\times height \). Here, base is \( 82.4\space m \), height is \( 35.9\space m \).

Step2: Calculate area

\( A=\frac{1}{2}\times82.4\times35.9 \)
First, \( 82.4\times35.9 = 82.4\times(30 + 5+ 0.9)=82.4\times30+82.4\times5 + 82.4\times0.9=2472+412+74.16 = 2958.16 \)
Then, \( \frac{1}{2}\times2958.16 = 1479.08\space m^2 \)

Part (b)

Step1: Analyze the wooded area

The wooded area can be considered as a trapezoid? Wait, no, looking at the diagram, maybe a combination? Wait, actually, the wooded area: let's see the bases and height. Wait, the lower part: we can use the formula for the area of a trapezoid? Wait, no, maybe a triangle? Wait, no, the diagram shows a quadrilateral? Wait, actually, let's check the dimensions. The base of the lower part: the horizontal length is \( 108.95\space m \), the other parallel side? Wait, no, maybe the area of the wooded part is a trapezoid with bases \( 24.5\space m \) and \( 108.95\space m \), and height \( 68.4\space m \)? Wait, no, wait the formula for the area of a trapezoid is \( A=\frac{(a + b)}{2}\times h \), where \( a \) and \( b \) are the two parallel sides, \( h \) is the height. Wait, but also, there's a small triangle? Wait, no, maybe I misread. Wait, the wooded area: let's re - examine. The lower part: the vertical side is \( 68.4\space m \), the horizontal sides: one is \( 24.5\space m \), the other is \( 108.95\space m \), and also, there's a small segment? Wait, no, maybe the area is the area of the trapezoid plus or minus? Wait, no, actually, the correct approach: the wooded area can be calculated as the area of a trapezoid with \( a = 24.5\space m \), \( b = 108.95\space m \), and \( h = 68.4\space m \), but also, there's a small triangle? Wait, no, the diagram shows that the wooded area is a trapezoid? Wait, no, let's check the formula again. Wait, maybe the wooded area is a trapezoid with bases \( 24.5\space m \) and \( 108.95\space m \), and height \( 68.4\space m \), but also, there's a small triangle with base \( 18.3\space m \)? No, maybe not. Wait, the correct way: the wooded area is a trapezoid with \( a = 24.5\space m \), \( b = 108.95\space m \), and height \( 68.4\space m \), and also, there's a small triangle? Wait, no, I think I made a mistake. Wait, the formula for the area of the wooded part: let's use the trapezoid formula \( A=\frac{(a + b)}{2}\times h \), where \( a = 24.5\), \( b = 108.95 \), \( h = 68.4 \)

Step2: Calculate trapezoid area

\( A=\frac{(24.5 + 108.95)}{2}\times68.4 \)
First, \( 24.5+108.95 = 133.45 \)
Then, \( \frac{133.45}{2}=66.725 \)
Then, \( 66.725\times68.4 = 66.725\times(60 + 8+ 0.4)=66.725\times60+66.725\times8+66.725\times0.4 = 4003.5+533.8+26.69 = 4563.99\space m^2 \)
Wait, but maybe there's another part? Wait, no, maybe the diagram has a small triangle with base \( 18.3\space m \) and height related? Wait, no, the problem says "la parcelle de terrain boisée au bas du croquis", so maybe the trapezoid is the main part. Wait, perhaps my initial approach is wrong. Wait, let's re - check the dimensions. The lower part: the horizontal length is \( 108.95\space m \), the upper horizontal segment is \( 24.5\space m \), and the vertical height is \( 68.4\space m \). So the area of a trapezoid is \( \frac{(a + b)}{2}\times h=\frac{(24.5 + 108.95)}{2}\times68.4=\frac{133.45}{2}\times68.4 = 66.725\times68.4 = 4563.99\space m^2 \approx4564\space m^2 \)

Answer:

(a):
The area of the triangular terrain by the stream is \( 1479.08\space m^2 \) (or approximately \( 1479.1\space m^2 \))