Answer :
To find the measure of angle between the points (-3,4), (1,4), and (1,8), we can use the dot product formula:dot product=AB⃗⋅BC⃗=(Bx−Ax)(Cx−Bx)+(By−Ay)(Cy−By)dot product=AB⋅BC=(Bx−Ax)(Cx−Bx)+(By−Ay)(Cy−By)And the formula for the magnitude of a vector:∣AB⃗∣=(Bx−Ax)2+(By−Ay)2∣AB∣=(Bx−Ax)2+(By−Ay)2Using these formulas, we can calculate the cosine of the angle between the vectors AB⃗AB and BC⃗BC as:cos(θ)=AB⃗⋅BC⃗∣AB⃗∣⋅∣BC⃗∣cos(θ)=∣AB∣⋅∣BC∣AB⋅BCFinally, we can find the measure of the angle θθ in degrees using the inverse cosine function:angle in degrees=cos−1(cos(θ))angle in degrees=cos−1(cos(θ))Substitute the coordinates into the formulas:AB⃗=(1−(−3),4−4)=(4,0)AB=(1−(−3),4−4)=(4,0)BC⃗=(1−1,8−4)=(0,4)BC=(1−1,8−4)=(0,4)dot product=(4⋅0)+(0⋅4)=0dot product=(4⋅0)+(0⋅4)=0∣AB⃗∣=(42+02)=16=4∣AB∣=(42+02)=16=4∣BC⃗∣=(02+42)=16=4∣BC∣=(02+42)=16=4cos(θ)=04⋅4=0cos(θ)=4⋅40=0θ=cos−1(0)=90∘θ=cos−1(0)=90∘So, the measure of the angle between the points is 90∘90∘
