Class 10 Maths · Triangles · NCERT Exercise 6.4 · Question 12

Triangles Hard

In a quadrilateral ABCD,

∠ A + ∠ D = 9 0^{\circ}

. Prove that

A C^{2} + B D^{2} = A D^{2} + B C^{2}

. [Hint: Produce AB and DC to meet at E.]

Answer

Proved: producing AB and DC to meet at E gives

∠ E = 9 0^{\circ}

; Pythagoras in the four right triangles shows both

A C^{2} + B D^{2}

and

A D^{2} + B C^{2}

equal

A E^{2} + B E^{2} + E C^{2} + E D^{2}

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