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) [tex]3\,in[/tex], c) [tex]4.5\,in[/tex], d) [tex]6.5\,in[/tex], e) [tex]10\,in[/tex], f) [tex]13.5\,in[/tex]
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:
[tex]y = \sqrt{l^{2}-x^{2}}[/tex] (1)
Where:
[tex]l[/tex] - Hypotenuse, in inches.
[tex]x[/tex] - Known leg, in inches.
[tex]y[/tex] - Unknown leg, in inches.
If we know that [tex]l = 11\,in[/tex] and [tex]x = 10\,in[/tex], then the length of the unknown leg is:
[tex]y = \sqrt{21}[/tex]
Since 4 is the least whole number closest to [tex]\sqrt{21}[/tex], 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:
[tex]x + y > l > x-y[/tex] (2)
Where:
[tex]x[/tex] - Greater side, in inches.
[tex]y[/tex] - Lesser side, in inches.
[tex]l[/tex] - Missing side, in inches.
If we know that [tex]x = 8\,in[/tex] and [tex]y = 6\,in[/tex], then we have the following range of missing sides:
[tex]14\,in > l > 2\,in[/tex]
The following straws can be used to construct the triangle: b) [tex]3\,in[/tex], c) [tex]4.5\,in[/tex], d) [tex]6.5\,in[/tex], e) [tex]10\,in[/tex], f) [tex]13.5\,in[/tex]
Exercise 4:
Let check each pair to determine possible constructions by means of the inequality used in Exercise 3:
(i) [tex]x = 4\,in, y = 2\,in[/tex]
[tex]6\,in>l>2\,in[/tex]
Possible choices: 5 inches.
(ii) [tex]x = 5\,in, y = 2\,in[/tex]
[tex]7\,in > l > 3\,in[/tex]
Possible choices: 4 inches, 6 inches.
(iii) [tex]x = 6\,in, y = 2\,in[/tex]
[tex]8\,in > l > 4\,in[/tex]
Possible choices: 5 inches, 6 inches.
(iv) [tex]x = 10\,in, y = 2\,in[/tex]
[tex]12\,in > l > 8\,in[/tex]
Possible choices: None.
(v) [tex]x = 5\,in, y = 4\,in[/tex]
[tex]9\,in > l > 1\,in[/tex]
Possible choices: 2 inches, 6 inches.
(vi) [tex]x = 6\,in, y = 4\,in[/tex]
[tex]10\,in > l > 2\,in[/tex]
Possible choices: 5 inches.
(vii) [tex]x = 10\,in, y = 4\,in[/tex]
[tex]14\,in > l > 6\,in[/tex]
Possible choices: None.
(viii) [tex]x = 6\,in, y = 5\,in[/tex]
[tex]11\,in > l > 1\,in[/tex]
Possible choices: 2 inches, 4 inches, 10 inches.
(ix) [tex]x = 10\,in, y = 5\,in[/tex]
[tex]15\,in > l > 5\,in[/tex]
Possible choices: 6 inches.
(x) [tex]x = 10\,in, y = 6\,in[/tex]
[tex]16\,in > l > 4\,in[/tex]
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)
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Step-by-step explanation:
