Application 2: Proving Triangles Inside a Rectangle are Equal

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Application 2: Proving Triangles Inside a Rectangle are EqualLet’s draw a rectangle and its Diagonals. We created 4 smaller triangles. They don't look exactly the same shape – so they are not Congruent But the area formula proves they take up the same amount of space. Let’s see how Proof We know that Diagonals of a rectangle are equal & bisect each other So, AB = CD And, OD = OB = OA = OC = 𝟏/𝟐 AB = 𝟏/𝟐 CD Now, let’s consider Triangle 1 & Triangle 2 Drawing Height from point A We notice that AX ⊥ BD Let’s find Area for Triangle 1 and Triangle 2 Area of Triangle 1 Here, Height = AX Base = OD Thus, Area of Triangle 1 = 1/2 × Base × Height = 𝟏/𝟐 × AX × OD Area of Triangle 2 This is an obtuse triangle And it’s height from vertex A is AX , And base is OB Now, Height = AX Base = OB Thus, Area of Triangle 2 = 1/2 × Base × Height = 𝟏/𝟐 × AX × OB Since OB = OD = 𝟏/𝟐 × AX × OD = Area of Triangle 1 Thus, we proved Area of Triangle 1 = Area of Triangle 2 Similarly, we can prove Area of Triangle 1 = Area of Triangle 2 = Area of Triangle 3 = Area of Triangle 4 Thus, area of all 4 triangles are equal