Question

decide whether the numbers can represent the side lengths of triangle. if they can, then determine if the triangle is right, acute, or obtuse.

1. 5,12,13
2. 20,21,28
3. 15,39,36
4. 14,48,50
5. 10,6,8
6. 5,7,9
7. 8,9,10
8. 10,12,22
9. 16,30,34
10. 7,7,10
11. 3,7,8
12. 12,12,3

match the side lengths with the appropriate description.

1. 26,20,17

a. right triangle

1. 26,20,14

b. acute triangle

1. 26,10,24

c. obtuse triangle

1. 26,10,15

d. not a triangle
challenge! an architect is trying to determine if the house he is constructing is being built on a perfectly flat ground. to reach the top of the house, which is 30 feet high, he places his 35 - foot ladder 16 feet away from the base of the house. is the ground perfectly flat? if so, how do you know? if not, is the ground slanted downhill or uphill from the house.

Explanation:

Step1: Recall triangle - side - length rules

For three positive real numbers \(a\), \(b\), and \(c\) (where \(c\) is the longest side) to be the side - lengths of a triangle, the triangle inequality theorem must hold: \(a + b>c\). To determine if it is a right - triangle, use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). For an acute triangle, \(a^{2}+b^{2}>c^{2}\), and for an obtuse triangle, \(a^{2}+b^{2}

Step2: Analyze 13. \(26,20,17\)

First, check the triangle inequality: \(20 + 17=37>26\), \(20+26 = 46>17\), \(17 + 26=43>20\). Then, check the type of triangle. Calculate \(a^{2}+b^{2}=20^{2}+17^{2}=400 + 289 = 689\) and \(c^{2}=26^{2}=676\). Since \(a^{2}+b^{2}>c^{2}\), it is an acute triangle (b).

Step3: Analyze 14. \(26,20,14\)

Check the triangle inequality: \(20 + 14=34>26\), \(20+26 = 46>14\), \(14 + 26=40>20\). Calculate \(a^{2}+b^{2}=20^{2}+14^{2}=400+196 = 596\) and \(c^{2}=26^{2}=676\). Since \(a^{2}+b^{2}

Step4: Analyze 15. \(26,10,24\)

Check the triangle inequality: \(10 + 24=34>26\), \(10+26 = 36>24\), \(24 + 26=50>10\). Calculate \(a^{2}+b^{2}=10^{2}+24^{2}=100 + 576=676\) and \(c^{2}=26^{2}=676\). Since \(a^{2}+b^{2}=c^{2}\), it is a right - triangle (a).

Step5: Analyze 16. \(26,10,15\)

Check the triangle inequality: \(10+15 = 25<26\). So, it is not a triangle (d).

Answer:

1. b. Acute Triangle
2. c. Obtuse Triangle
3. a. Right Triangle
4. d. Not a Triangle