Question

1. if points m, n, and p are collinear with point n between points m and p, then which of the following is not true?

(1) ( mn + np = mp )
(2) ( np < mp + mn )
(3) ( mn < mp + np )
(4) ( mp < mn + np )

1. two sides of a triangle have lengths of 13 inches and 18 inches. which of the following could not be the length of its third side?

(1) 6 inches
(2) 8 inches
(3) 13 inches
(4) 33 inches

1. two sides of a triangle have lengths of 8.25 meters and 14.75 meters. which of the following could be the length of its third side?

(1) 4.5 meters
(2) 6.25 meters
(3) 18.3 meters
(4) 23.5 meters

1. in ( \triangle efg ), ( ef = 22 ) and ( fg = 17 ). which of the following represents all possible values of ( ge )?

(1) ( 17 < ge < 22 )
(2) ( 5 < ge < 39 )
(3) ( 22 < ge < 39 )
(4) ( 0 < ge < 17 )

1. which of the following sets of numbers could not represent the side lengths of a triangle?

(1) ( {11, 17, 25} )
(2) ( {8, 8, 8} )
(3) ( {5, 8, 15} )
(4) ( {6, 10, 10} )

Explanation:

1.

Step1: Analyze collinear points property

Since \(M\), \(N\), \(P\) are collinear and \(N\) is between \(M\) and \(P\), by the segment - addition postulate \(MN + NP=MP\).

Step2: Check each option

• Option (1): \(MN + NP = MP\) (True).
• Option (2): \(NP=MP - MN\), so \(NP
• Option (3): \(MN=MP - NP\), so \(MN
• Option (4): Since \(MN + NP = MP\), \(MP

2.

Step1: Use triangle - inequality theorem

For a triangle with side lengths \(a = 13\) and \(b = 18\), the third - side \(c\) satisfies \(|a - b|\(|13 - 18|=5\) and \(13 + 18 = 31\). So \(5

Step2: Check each option

• Option (1): \(6\) satisfies \(5<6<31\).
• Option (2): \(8\) satisfies \(5<8<31\).
• Option (3): \(13\) satisfies \(5<13<31\).
• Option (4): \(33\) does not satisfy \(5<33<31\).

3.

Step1: Apply triangle - inequality theorem

For a triangle with side lengths \(a = 8.25\) and \(b = 14.75\), the third - side \(c\) satisfies \(|a - b|\(|8.25 - 14.75|=6.5\) and \(8.25+14.75 = 23\). So \(6.5

Step2: Check each option

• Option (1): \(4.5<6.5\) (Does not satisfy).
• Option (2): \(6.5<6.25<23\) (Does not satisfy).
• Option (3): \(6.5<18.3<23\) (Satisfies).
• Option (4): \(23.5>23\) (Does not satisfy).

4.

Step1: Use triangle - inequality theorem

In \(\triangle EFG\) with \(EF = 22\) and \(FG = 17\), for side \(GE\), \(|EF - FG|\(|22 - 17| = 5\) and \(22+17 = 39\). So \(5

5.

Step1: Apply triangle - inequality theorem

For a set of numbers \(\{a,b,c\}\) to be side lengths of a triangle, \(a + b>c\), \(a + c>b\), and \(b + c>a\).

• For \(\{11,17,25\}\): \(11+17 = 28>25\), \(11 + 25=36>17\), \(17+25 = 42>11\).
• For \(\{8,8,8\}\): \(8 + 8>8\).
• For \(\{5,8,15\}\): \(5+8=13<15\).
• For \(\{6,10,10\}\): \(6 + 10>10\), \(6+10>10\), \(10 + 10>6\).

Answer:

1. (4)
2. (4)
3. (3)
4. (2)
5. (3)