Answer:
Exercise 1: The length of the unknown leg is 4 inches.
Exercise 3: The following straws can be used to construct the triangle: b)
, c)
, d)
, e)
, f) ![13.5\,in](https://tex.z-dn.net/?f=13.5%5C%2Cin)
Exercise 4: Possible options of this exercise: 1) (2, 4, 5), 2) (4, 5, 6), 3) (5, 6, 10), 4) (1, 2, 11), 5) (1, 4, 11), 6) (1, 10, 11)
Step-by-step explanation:
Exercise 1:
Let suppose that triangle represented in the figure is a right triangle, the length of the missing leg is determined by Pythagorean Theorem:
(1)
Where:
- Hypotenuse, in inches.
- Known leg, in inches.
- Unknown leg, in inches.
If we know that
and
, then the length of the unknown leg is:
![y = \sqrt{21}](https://tex.z-dn.net/?f=y%20%3D%20%5Csqrt%7B21%7D)
Since 4 is the least whole number closest to
, then we conclude that the length of the unknown leg is 4 inches.
Exercise 3:
The range of possible lengths for the missing side of the triangle is represented by the following simultaneous inequality:
(2)
Where:
- Greater side, in inches.
- Lesser side, in inches.
- Missing side, in inches.
If we know that
and
, then we have the following range of missing sides:
![14\,in > l > 2\,in](https://tex.z-dn.net/?f=14%5C%2Cin%20%3E%20l%20%3E%202%5C%2Cin)
The following straws can be used to construct the triangle: b)
, c)
, d)
, e)
, f) ![13.5\,in](https://tex.z-dn.net/?f=13.5%5C%2Cin)
Exercise 4:
Let check each pair to determine possible constructions by means of the inequality used in Exercise 3:
(i) ![x = 4\,in, y = 2\,in](https://tex.z-dn.net/?f=x%20%3D%204%5C%2Cin%2C%20y%20%3D%202%5C%2Cin)
![6\,in>l>2\,in](https://tex.z-dn.net/?f=6%5C%2Cin%3El%3E2%5C%2Cin)
Possible choices: 5 inches.
(ii) ![x = 5\,in, y = 2\,in](https://tex.z-dn.net/?f=x%20%3D%205%5C%2Cin%2C%20y%20%3D%202%5C%2Cin)
![7\,in > l > 3\,in](https://tex.z-dn.net/?f=7%5C%2Cin%20%3E%20l%20%3E%203%5C%2Cin)
Possible choices: 4 inches, 6 inches.
(iii) ![x = 6\,in, y = 2\,in](https://tex.z-dn.net/?f=x%20%3D%206%5C%2Cin%2C%20y%20%3D%202%5C%2Cin)
![8\,in > l > 4\,in](https://tex.z-dn.net/?f=8%5C%2Cin%20%3E%20l%20%3E%204%5C%2Cin)
Possible choices: 5 inches, 6 inches.
(iv) ![x = 10\,in, y = 2\,in](https://tex.z-dn.net/?f=x%20%3D%2010%5C%2Cin%2C%20y%20%3D%202%5C%2Cin)
![12\,in > l > 8\,in](https://tex.z-dn.net/?f=12%5C%2Cin%20%3E%20l%20%3E%208%5C%2Cin)
Possible choices: None.
(v) ![x = 5\,in, y = 4\,in](https://tex.z-dn.net/?f=x%20%3D%205%5C%2Cin%2C%20y%20%3D%204%5C%2Cin)
![9\,in > l > 1\,in](https://tex.z-dn.net/?f=9%5C%2Cin%20%3E%20l%20%3E%201%5C%2Cin)
Possible choices: 2 inches, 6 inches.
(vi) ![x = 6\,in, y = 4\,in](https://tex.z-dn.net/?f=x%20%3D%206%5C%2Cin%2C%20y%20%3D%204%5C%2Cin)
![10\,in > l > 2\,in](https://tex.z-dn.net/?f=10%5C%2Cin%20%3E%20l%20%3E%202%5C%2Cin)
Possible choices: 5 inches.
(vii) ![x = 10\,in, y = 4\,in](https://tex.z-dn.net/?f=x%20%3D%2010%5C%2Cin%2C%20y%20%3D%204%5C%2Cin)
![14\,in > l > 6\,in](https://tex.z-dn.net/?f=14%5C%2Cin%20%3E%20l%20%3E%206%5C%2Cin)
Possible choices: None.
(viii) ![x = 6\,in, y = 5\,in](https://tex.z-dn.net/?f=x%20%3D%206%5C%2Cin%2C%20y%20%3D%205%5C%2Cin)
![11\,in > l > 1\,in](https://tex.z-dn.net/?f=11%5C%2Cin%20%3E%20l%20%3E%201%5C%2Cin)
Possible choices: 2 inches, 4 inches, 10 inches.
(ix) ![x = 10\,in, y = 5\,in](https://tex.z-dn.net/?f=x%20%3D%2010%5C%2Cin%2C%20y%20%3D%205%5C%2Cin)
![15\,in > l > 5\,in](https://tex.z-dn.net/?f=15%5C%2Cin%20%3E%20l%20%3E%205%5C%2Cin)
Possible choices: 6 inches.
(x) ![x = 10\,in, y = 6\,in](https://tex.z-dn.net/?f=x%20%3D%2010%5C%2Cin%2C%20y%20%3D%206%5C%2Cin)
![16\,in > l > 4\,in](https://tex.z-dn.net/?f=16%5C%2Cin%20%3E%20l%20%3E%204%5C%2Cin)
Possible choices: 5 inches.
Possible options of this exercise: 1) (2, 4, 5), 2) (4, 5, 6), 3) (5, 6, 10), 4) (1, 2, 11), 5) (1, 4, 11), 6) (1, 10, 11)