Answer:
[tex]\alpha=69.28^o[/tex]
[tex]\beta=61.26^o[/tex]
[tex]\gamma=49.46^o[/tex]
[tex]A=227980.26\ m^2[/tex]
Step-by-step explanation:
Triangle Solving
If we had a triangle will its three sides of known length, we can solve for the rest of the parameters of the triangle, i.e. the area, perimeter and internal angles.
The three circles have diameters 900 m, 700 m and 600 m and are tangent to each other externally. The distances from their centers (where houses are located) are the sum of each pair of the radius of the circles. Thus, the sides of the triangle are
[tex]x=450+350=800[/tex]
[tex]y=450+300=750[/tex]
[tex]z=350+300=650[/tex]
The internal angles can be computed by using the cosine's law
[tex]x^2=y^2+z^2-2yzcos\alpha[/tex]
[tex]y^2=x^2+z^2-2xzcos\beta[/tex]
[tex]z^2=x^2+y^2-2xycos\gamma[/tex]
Where \alpha, \beta and \gamma are the opposite angles to x, y and z respectively. Solving for each one of them:
[tex]\displaystyle cos\alpha=\frac{y^2+z^2-x^2}{2yz}[/tex]
[tex]\displaystyle cos\alpha=\frac{750^2+650^2-800^2}{2\cdot 750\cdot 650}[/tex]
[tex]cos\alpha=0.3538[/tex]
[tex]\alpha=69.28^o[/tex]
Similarly
[tex]\displaystyle cos\beta=\frac{x^2+z^2-y^2}{2xz}[/tex]
[tex]\displaystyle cos\beta=\frac{800^2+650^2-750^2}{2\cdot 800\cdot 650}[/tex]
[tex]cos\beta=0.4808[/tex]
[tex]\beta=61.26^o[/tex]
The other angle is computed now
[tex]\displaystyle cos\gamma=\frac{x^2+y^2-z^2}{2xy}[/tex]
[tex]\displaystyle cos\gamma=\frac{800^2+750^2-650^2}{2\cdot 800\cdot 750}[/tex]
[tex]cos\gamma=0.65[/tex]
[tex]\gamma=49.46^o[/tex]
The area can be found by
[tex]\displaystyle A=\frac{1}{2}x.y.sin\gamma[/tex]
[tex]\displaystyle A=\frac{1}{2}\cdot 800\cdot 750\cdot sin49.46^o[/tex]
[tex]A=227980.26\ m^2[/tex]
The angles of the triangle connecting the houses are α ≈ 69.277°, β ≈ 61.264° and γ ≈ 49.458°, respectively.
The area of the triangle is approximately 227980.262 square meters.
Three circles that are externally tangent among them form a triangle, each side of the triangle is colinear to the corresponding line of tangency. First, we need to determine each angle by using the law of the cosine, whose formula is shown below:
[tex]\alpha = \cos^{-1}\left[\frac{(750)^{2}+(650)^{2}-(800)^{2}}{2\cdot (750)\cdot (650)} \right][/tex]
[tex]\alpha \approx 69.277^{\circ}[/tex]
[tex]\beta = \cos^{-1}\left[\frac{(650)^{2}+(800)^{2}-(750)^{2}}{2\cdot (650)\cdot (800)} \right][/tex]
[tex]\beta \approx 61.264^{\circ}[/tex]
[tex]\gamma = \cos^{-1}\left[\frac{(750)^{2}+(800)^{2}-(650)^{2}}{2\cdot (750)\cdot (800)} \right][/tex]
[tex]\gamma \approx 49.458^{\circ}[/tex]
The angles of the triangle connecting the houses are α ≈ 69.277°, β ≈ 61.264° and γ ≈ 49.458°, respectively. [tex]\blacksquare[/tex]
The area of the triangle is determine by the Heron's formula:
[tex]s = \frac{650+750+800}{2}[/tex]
[tex]s = 1100[/tex]
[tex]A = \sqrt{(1100)\cdot (1100-650)\cdot (1100-750)\cdot (1100-800)}[/tex]
[tex]A \approx 227980.262\,m^{2}[/tex]
The area of the triangle is approximately 227980.262 square meters. [tex]\blacksquare[/tex]
To learn more on triangles, we kindly invite to check this verified question: https://brainly.com/question/2773823
