Math  /  Geometry

QuestionFind the area and perimeter of DEF\triangle D E F.

Studdy Solution

STEP 1

1. We are given a triangle DEF \triangle DEF on a coordinate plane.
2. The vertices of the triangle are D(3,3) D(-3, 3) , E(3,3) E(3, 3) , and F(0,3) F(0, -3) .
3. We need to find both the area and the perimeter of the triangle.

STEP 2

1. Determine the lengths of the sides of the triangle using the distance formula.
2. Calculate the area of the triangle using the appropriate method.
3. Calculate the perimeter of the triangle by summing the lengths of its sides.

STEP 3

To find the lengths of the sides of the triangle, we will use the distance formula. The distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:
d=(x2x1)2+(y2y1)2 d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

STEP 4

Calculate the length of side DE DE :
DE=(3(3))2+(33)2 DE = \sqrt{(3 - (-3))^2 + (3 - 3)^2} =(3+3)2+02 = \sqrt{(3 + 3)^2 + 0^2} =62 = \sqrt{6^2} =6 = 6

STEP 5

Calculate the length of side EF EF :
EF=(03)2+(33)2 EF = \sqrt{(0 - 3)^2 + (-3 - 3)^2} =(3)2+(6)2 = \sqrt{(-3)^2 + (-6)^2} =9+36 = \sqrt{9 + 36} =45 = \sqrt{45} =35 = 3\sqrt{5}

STEP 6

Calculate the length of side DF DF :
DF=(0(3))2+(33)2 DF = \sqrt{(0 - (-3))^2 + (-3 - 3)^2} =(0+3)2+(6)2 = \sqrt{(0 + 3)^2 + (-6)^2} =32+36 = \sqrt{3^2 + 36} =9+36 = \sqrt{9 + 36} =45 = \sqrt{45} =35 = 3\sqrt{5}

STEP 7

To find the area of the triangle, we will use the formula for the area of a triangle given vertices (x1,y1)(x_1, y_1), (x2,y2)(x_2, y_2), (x3,y3)(x_3, y_3):
Area=12x1(y2y3)+x2(y3y1)+x3(y1y2) \text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|

STEP 8

Substitute the coordinates of D(3,3) D(-3, 3) , E(3,3) E(3, 3) , and F(0,3) F(0, -3) into the area formula:
Area=123(3(3))+3(33)+0(33) \text{Area} = \frac{1}{2} \left| -3(3 - (-3)) + 3(-3 - 3) + 0(3 - 3) \right| =123(6)+3(6)+0 = \frac{1}{2} \left| -3(6) + 3(-6) + 0 \right| =121818 = \frac{1}{2} \left| -18 - 18 \right| =12×36 = \frac{1}{2} \times 36 =18 = 18

STEP 9

To find the perimeter of the triangle, sum the lengths of its sides:
Perimeter=DE+EF+DF \text{Perimeter} = DE + EF + DF =6+35+35 = 6 + 3\sqrt{5} + 3\sqrt{5} =6+65 = 6 + 6\sqrt{5}
The area of DEF \triangle DEF is 18 18 square units, and the perimeter is 6+65 6 + 6\sqrt{5} units.

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