Question

use the 5 by 5 square lattice shown, with the line segment ab. list all the points c of the lattice that make ∠bac the following kinds of angle. do not include points a or b in any of your answers
(a) a right angle
2,7,12,22
(type a whole number. use a comma to separate answers as needed.)
(b) an acute angle
3,4,5,6,8,9,10,13,14,15,23,24,25
(type a whole number. use a comma to separate answers as needed.)
(c) an obtuse angle

(type a whole number. use a comma to separate answers as needed.)

Explanation:

Step1: Identify right angle points

We use the property that for $\angle BAC$ to be right, vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are perpendicular (dot product = 0). $\overrightarrow{AB}=(2,0)$. For each lattice point $C(x,y)$, $(x-1,y-2)\cdot(2,0)=2(x-1)=0 \implies x=1$. Valid points (excluding A,B): 2,12,22

Step2: Identify acute angle points

$\angle BAC$ is acute if the dot product of $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is positive, and the angle between them is less than 90°. Calculate for all points not A/B/right angle points: 3,4,5,8,9,10,13,14,15,21,23,24,25

Step3: Identify obtuse angle points

$\angle BAC$ is obtuse if the dot product of $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is negative, or the angle between them is greater than 90°. Remaining valid points (excluding A/B/right/acute): 1,6,7,11,16,17,18,19,20

Answer:

a) 2, 12, 22
b) 3, 4, 5, 8, 9, 10, 13, 14, 15, 21, 23, 24, 25
c) 1, 6, 7, 11, 16, 17, 18, 19, 20