See proof below
Step-by-step explanation:
Assume triangle ABC to have vertices at;
A(2,-1), B(2,-7) and C(6,-7)
D is midpoint of BC, thus D is at (4,-7)
The P and Q, are lying on side AB and AC, hence assume P is at (2,-4) and Q is at (4,-4) such at DP is parallel to QA
Plot the points on a graph tool and join the points to view the sketch.
To prove area of triangle CPQ is 1/4 area of ABC will be;
Find area ABC and CPQ then compare the areas.
Apply the distance formula to find the length of sides of the triangles then find the areas.
The distance formula is;
[tex]d=\sqrt{x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Length of side AB from the sketch is;
[tex]AB=\sqrt{(2-2)^2+(-7--1)^2} =\sqrt{-6^2} =\sqrt{36} =6units[/tex]
Length of side BC will be;
[tex]BC=\sqrt{(-7--7)^2+(6-2)^2} =\sqrt{4^2} =\sqrt{16} =4units[/tex]
Thus area of triangle ABC will be;
1/2 *base length*height ------because it is a right-triangle
1/2*4*6=12 square units
Find the lengths of all sides of triangle CPQ
Length of side PQ is half that of side BC thus PQ=2 units
Length of side PC is;
[tex]PC=\sqrt{(6-2)^2+(-7--4)^2} =\sqrt{4^2+-3^2} =\sqrt{16+9} =\sqrt{25} =5units[/tex]
Length of side QC will be;
[tex]QC=\sqrt{(6-4)^2+(-7--4)^2} =\sqrt{2^2+-3^3} =\sqrt{4+9} =\sqrt{13}[/tex]
QC= √13 = 3.6 units
Find area of triangle CPQ given all sides by applying the Heron's formula for area of triangle which is;
A=√s(s-a)(s-b)(s-c) where;
A=area of the triangle
s= half the perimeter of the triangle
a=side PQ = 2 units
b=side PC = 5 units
c= side QC = 3.6 units
Finding the perimeter of triangle CPQ will be;
P=sum of all sides
P=2+5+3.6 =10.6 units
s=10.6/2 = 5.3
Area of the triangle CPQ will be;
[tex]A=\sqrt{5.3(5.3-2)(5.3-5)(5.3-3.6)} \\A=\sqrt{5.3(3.3)(0.3)(1.7)} \\A=\sqrt{8.9} =2.98[/tex]
A=3.0 (1 decimal place)
Compare the areas;
Area of triangle ABC=12 square units
Area of triangle CPQ = 3 square units
Area of triangle CPQ / Area of triangle ABC = 3/12 =1/4
Thus you have proved that area of triangle CPQ is 1/4 th area of triangle ABC because 1/4 *12 =3
Learn More
Area of a triangle ;https://brainly.com/question/14869984
The Heron's formula : https://brainly.com/question/10713495
Keywords: midpoint, triangle, sides, parallel, prove , area, equal
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