Question

1. a triangular pendant has sides of length 23 mm and 31 mm. what are the possible lengths (in whole mm) for the third side? a 8 mm or less b between 9 mm and 53 mm, inclusive c 54 mm or more d not possible 14. select all the true statements. a. a circumcenter is always inside of its triangle. b. an incenter is always inside of its triangle. c. an orthocenter is always inside of its triangle. d. a centroid is always inside of its triangle. e. a circumcenter, orthocenter, incenter, or centroid can be either inside or outside of its triangle.

Explanation:

Question 13

Step1: Recall triangle inequality theorem

The triangle inequality theorem states that the length of any side of a triangle must be greater than the absolute difference of the other two sides and less than the sum of the other two sides. Let the third side be \( x \).

Step2: Calculate the range

First, find the sum of the two given sides: \( 23 + 31 = 54 \). Then, find the absolute difference: \( |31 - 23| = 8 \). So, the inequality for the third side \( x \) is \( 8 < x < 54 \). Since we need whole mm, \( x \) is between 9 mm and 53 mm (inclusive, because when \( x = 9 \), \( 23 + 9>31 \) (23 + 9 = 32>31), \( 31 + 9>23 \) (40>23), \( 23 + 31>9 \) (54>9); when \( x = 53 \), \( 23+53 > 31 \) (76>31), \( 31 + 53>23 \) (84>23), \( 23+31>53 \) (54>53)).

• Statement A: The circumcenter is the intersection of perpendicular bisectors. In an obtuse triangle, the circumcenter lies outside the triangle. So, A is false.
• Statement B: The incenter is the intersection of angle bisectors. Since angle bisectors always lie inside the triangle, the incenter is always inside. So, B is true.
• Statement C: The orthocenter is the intersection of altitudes. In an obtuse triangle, the orthocenter lies outside the triangle. So, C is false.
• Statement D: The centroid is the intersection of medians. Medians always lie inside the triangle, so the centroid is always inside. So, D is true.
• Statement E: As we saw, incenter and centroid are always inside, while circumcenter and orthocenter can be outside (in obtuse triangles) or inside (in acute triangles). So, E is false.

Answer:

B. between 9 mm and 53 mm, inclusive

Question 14