Question

unit 7 right triangles and trigonometry homework 3 similar right triangles and Geometric Mean, pls help.

Answer 1

The results are listed below:

11) The value of $x$ is approximately 32.404 units.

12) The value of $x$ is approximately 10.392 units.

13) The value of $x$ is approximately 15.875 units.

14) The value of $x$ is approximately 22.450 units.

15) The value of $x$ is 11.571 units.

16) The value of $x$ is 17 units.

17) The value of $x$ is approximately 21.333 units.

18) The value of $x$ is 24.857 units.

19) The solution of this system is: $x \approx 16.155$, $y \approx 6.462$, $z\approx 2.4$.

20) The solution of this system is: $x = 37.5$, $y = 18$, $z = 22.5$.

How to calculate missing sides in right triangles

In this question we should apply Pythagorean theorem and algebraic methods to find the missing lengths. Now we proceed to the procedure for each case:

Figure 11

We construct the system of equations and find its solution:

$-x^{2}+y^{2} = 40^{2}$ (1)

$y^{2}+z^{2} = 62^{2}$ (2)

$-x^{2}+z^{2} = 12^{2}$ (3)

The solution of this system is: $x \approx 32.404$, $y\approx 51.478$, $z\approx 34.554$. $\blacksquare$

Figure 12

We construct the system of equations and find its solution:

$-x^{2} + y^{2} = 3^{2}$ (1)

$-x^{2}+z^{2} = 36^{2}$ (2)

$y^{2}+z^{2} = 39^{2}$ (3)

The solution of this system is: $x \approx 10.392$, $y \approx 10.817$, $z \approx 37.470$. $\blacksquare$

Figure 13

We construct the system of equations and find its solution:

$x^{2}-y^{2} = 14^{2}$ (1)

$-y^{2}+z^{2} = 4^{2}$ (2)

$x^{2}+z^{2} = 18^{2}$ (3)

The solution of this system is: $x \approx 15.875$, $y\approx 7.483$, $z \approx 8.485$. $\blacksquare$

Figure 14

We construct the system of equations and find its solution:

$x^{2}-y^{2} = 8^{2}$ (1)

$-y^{2}+z^{2} = 8^{2}$ (2)

$x^{2}+z^{2} = 63^{2}$ (3)

The solution of this system is: $x \approx 22.450$, $y \approx 20.976$, $z \approx 58.865$. $\blacksquare$

Figure 15

We construct the system of equations and find its solution:

$-x^{2}+y^{2} = 18^{2}$ (1)

$z^{2} = 28^{2}+18^{2}$ (2)

$z^{2}+y^{2} = (x+28)^{2}$ (3)

The solution of this system is: $x = 11.571$, $y \approx 21.398$, $z\approx 33.287$. $\blacksquare$

Figure 16

We construct the system of equations and find its solution:

$y^{2} = 20^{2}-8^{2}$ (1)

$x^{2}+y^{2}-z^{2} = 0$ (2)

$(x+8)^{2}-z^{2}= 20^{2}$ (3)

The solution of this system is: $x = 17$, $y \approx 18.330$, $z = 25$. $\blacksquare$

Figure 17

We construct the system of equations and find its solution:

$y^{2} = 16^{2}-12^{2}$ (1)

$(x-12)^{2}+y^{2}-z^{2} = 0$ (2)

$x^{2}-z^{2} = 16^{2}$ (3)

The solution of this system is: $x \approx 21.333$, $y\approx 10.583$, $z \approx 14.110$. $\blacksquare$

Figure 18

We construct the system of equations and find its solution:

$-(x-21)^{2}+y^{2} = 9^{2}$ (1)

$z^{2} = 21^{2}+9^{2}$ (2)

$-x^{2}+y^{2}+z^{2} = 0$ (3)

The solution of this system is: $x = 24.857, y \approx 9.791, z \approx 22.847$. $\blacksquare$

Figure 19

We construct the system of equations and find its solution:

$y^{2}-z^{2} = 6^{2}$ (1)

$x^{2} = 6^{2}+15^{2}$ (2)

$x^{2}+y^{2}-(15+z)^{2} = 0$ (3)

The solution of this system is: $x \approx 16.155$, $y \approx 6.462$, $z\approx 2.4$. $\blacksquare$

Figure 20

We construct the system of equations and find its solution:

$y^{2} = 30^{2}-24^{2}$ (1)

$(x-24)^{2}+y^{2}-z^{2} = 0$ (2)

$x^{2}-z^{2} = 30^{2}$ (3)

The solution of this system is: $x = 37.5$, $y = 18$, $z = 22.5$. $\blacksquare$

To learn more on right triangles, we kindly invite to check this verified question: https://brainly.com/question/7894175