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QuestionFind the perimeter of triangle with vertices at P(6,4)P(6,4), Q(−3,1)Q(-3,1), and R(9,−5)R(9,-5) in surd form.

Studdy Solution

STEP 1

Assumptions1. The vertices of the triangle are (6,4)(6,4), Q(−3,1)Q(-3,1) and R(9,−5)R(9,-5). . The perimeter of a triangle is the sum of the lengths of all its sides.

STEP 2

We need to find the lengths of the sides of the triangle. The distance between two points (x1,y1)(x1, y1) and (x2,y2)(x2, y2) in a2 plane is given by the formulad=(x2−x1)2+(y2−y1)2d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

STEP 3

Let's find the length of side QQ first. Plug in the coordinates of points and $Q$ into the distance formulaQ = \sqrt{((-3 -6)^2 + (1 -)^2)}$$

STEP 4

implify the expression inside the square root to get the length of QQQ=((−9)2+(−3)2)=(81+9)=90Q = \sqrt{((-9)^2 + (-3)^2)} = \sqrt{(81 +9)} = \sqrt{90}

STEP 5

Next, let's find the length of side QRQR. Plug in the coordinates of points QQ and RR into the distance formulaQR=((9−(−3))2+((−5)−1)2)QR = \sqrt{((9 - (-3))^2 + ((-5) -1)^2)}

STEP 6

implify the expression inside the square root to get the length of QRQRQR=(122+(−6)2)=(144+36)=180QR = \sqrt{(12^2 + (-6)^2)} = \sqrt{(144 +36)} = \sqrt{180}

STEP 7

Finally, let's find the length of side RPRP. Plug in the coordinates of points RR and intothedistanceformula into the distance formulaRP = \sqrt{((6 -9)^2 + (4 - (-5))^2)}$$

STEP 8

implify the expression inside the square root to get the length of RPRPRP=((−3)2+2)=(+81)=90RP = \sqrt{((-3)^2 +^2)} = \sqrt{( +81)} = \sqrt{90}

STEP 9

Now that we have the lengths of all sides, we can find the perimeter of the triangle. The perimeter is the sum of the lengths of all sidesPerimeter=PQ+QR+RPPerimeter = PQ + QR + RP

STEP 10

Plug in the values for QQ, QRQR and RPRP to calculate the perimeterPerimeter=90+180+90Perimeter = \sqrt{90} + \sqrt{180} + \sqrt{90}

STEP 11

implify the expression to get the perimeter of the trianglePerimeter=90+180Perimeter =\sqrt{90} + \sqrt{180}The perimeter of the triangle is 90+180\sqrt{90} + \sqrt{180}.

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