Find the area and perimeter of the figure determined by the points. (−5, 5)​, (0, 1)​, (4, 6)

Question

Find the area and perimeter of the figure determined by the points. (−5, 5)​, (0, 1)​, (4, 6)

​(Simplify your answer. Type an exact​ answer, using radicals as​ needed.)

Answer 1

Let A= the 1st point (-5,5) and also let A= Angle BAC,

B= 2nd point,(0,1) and B also = Angle ABC

C=3rd point (4, 6) and also C= Angle ACB

use the distance formula

d^2 = sum of squares of the difference in x coordinates and difference in y coordinates

AB = sqr(5^3 + 4^2) = sqr(25+16)=sqr41

BC= sqr(25+16)= sqr41

CA= sqr(9^2 + 1^2) = sqr(81+1) = sqr82= sqr2(sqr41)

use Law of Cosines

AB^2 = BC^2 + CA^ -2(BC)(CA)CosC where C=angle opposite side AB = angle opposite BC = A

41 = 41 +82 - 2sqr((41)(82))CosC

2(41)sqr2CosC = 82

CosC =1/sqr2

C=Cos^-1(sqr2/2) = 45 degrees.= A

B-180-2(45) = 90 degrees

It's a right triangle with hypotenuse = sqr2(sqr41) and other two sides = sqr41

Area = base times height divided by 2

= sqr41(sqr41)/2 = 41/2 = 20.5

Angle opposite side CA =B= 180 -C-A It's an isosceles right triangle. Two sides are the same length and two angles are equal A=C

the third angle is 180 minus the sum of the other two sides: B= 180-2C

construct two right triangle cutting the triangle down the middle, then the base is half of CA and the height = ABsinC

Area of the triangle = height times base/2 = hb/2 = ABsinC(CA)/2

or use

Heron's Formula for when you know all 3 sides of a triangle

perimeter= P = AB+BA+CA

perimeter = P = (2+sqr2)(sqr41)

half the perimpeter= P/2 = p=(1/2)((sqr2+2)(sqr41)

Area = sqr((p)(p-sqr41)(p-sqr41)(p-sqr82) where p= 1/2 the perimeter

There's no guarantee there are no arithmetic mistakes above somewhere, but that's the general method or several methods to find the area. It may help to plot the points, connect them, and draw the triangle. With even a rough sketch, you should more easily see it's a right triangle from the very beginning. Then to find area just square one of the smaller sides and divide by 2. ((sqr41)^2)/2 = 41/2 = 20.5 = 20 1/2 square units, Perimeter = sum of the 3 sides: sqr41 + sqr41 + sqr82 = 2sqr41 + sqr2(sqr41) = (2+sqr2)sqr41

Or you could find the equations of the lines that connect the 3 points, calculate their slopes, and if two of the slopes are negative inverses, then it's a right triangle. -4/5 and 5/4 are the slopes of two of the sides of the triangle.