Can someone help me on this please

∠B = 68.20°

Step-by-step explanation:

Since this is a right triangle, we can use the trigonometry ratios to solve. Remember them using the acronym "SohCahToa".

sinθ = opposite/hypotenuse

cosθ = adjacent/hypotenuse

tanθ = opposite/adjacent

θ means the angle of reference, or angle you are talking about.

Hypotenuse is the longest side. Adjacent side touches θ and is not the hypotenuse. The opposite side does not touch θ.

In this problem, θ = ∠B.

The side we know are AC = 5 and BC = 2.

To ∠B, BC is adjacent. AC is opposite.

The trig. ratio with opposite and adjacent is:

tanθ = opposite/adjacent

Use this ratio formula and solve for ∠B.

tanθ = opposite/adjacent

tanB = AC/BC                   Insert variables from the diagram

tanB = 5/2                     Substitute known values

B = tan⁻¹(5/2)                    Isolate "B"

B = tan⁻¹(2.5)                    Simplified fraction

B = 68.19859051°             Unrounded answer on your calculator

B ≈ 68.20°                 Rounded to the nearest hundredth

How to round:

Focus on one more digit than you are rounding to:

Hundredth is the 2nd decimal digit. One over to the right is the third:

68.198 5905

The third decimal digit determines if you round up or down.

Round up if: 5 or greater

Round down if: 4 or less

8 is 5 or greater, so round up by increasing the hundredth digit by one. Drop the third decimal digit.

68.19

+0.01

68.20

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!