Question

In a diamond ABCD, we draw the segment BN perpendicular to the segment AD, the segment BM perpendicular to CD, the segment DR perpendicular to AB and the segment DQ perpendicular to BC. The perpendiculars BN and DR intersect at point E and the perpendiculars BM and DQ intersect at point F. Demonstrate using Euclidean geometry that BEDF is a rhombus and that its angles are isometric to the angles of rhombus ABCD.

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Answer to a math question In a diamond ABCD, we draw the segment BN perpendicular to the segment AD, the segment BM perpendicular to CD, the segment DR perpendicular to AB and the segment DQ perpendicular to BC. The perpendiculars BN and DR intersect at point E and the perpendiculars BM and DQ intersect at point F. Demonstrate using Euclidean geometry that BEDF is a rhombus and that its angles are isometric to the angles of rhombus ABCD.

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Esmeralda
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Pour démontrer que BEDF est un losange et que ses angles sont isométriques aux angles du losange ABCD, nous pouvons utiliser la géométrie euclidienne et les informations données. Commençons par analyser la figure donnée et les propriétés des lignes et segments mentionnés. 1. Nous avons une figure ABCD en forme de losange, où AD et BC sont des perpendiculaires égales au segment de droite AB. 2. Le segment BN est perpendiculaire à AD et le segment DR est perpendiculaire à AB. Ces perpendiculaires se coupent au point E. 3. Le segment BM est perpendiculaire à CD et le segment DQ est perpendiculaire à BC. Ces perpendiculaires se coupent au point F. Montrons maintenant que BEDF est un losange : 1. Pour montrer que BEDF est un losange, nous devons prouver que les quatre côtés sont congrus. un. Puisque BN et DR sont respectivement perpendiculaires à AD et AB, ils sont tous deux médiateurs perpendiculaires de AB. Cela signifie que le point E se trouve sur la médiatrice de AB et que la distance de E à A et B est égale. De même, le point F se trouve sur la médiatrice de AB et la distance de F à A et B est égale. b. On peut donc conclure que BE = EA = DF = FA. 2. Pour montrer que les angles de BEDF sont isométriques aux angles de ABCD, nous devons prouver que les angles correspondants sont congrus. un. Puisque BN est perpendiculaire à AD et BM est perpendiculaire à CD, l’angle BNE et l’angle BMF sont tous deux des angles droits. b. De même, puisque DR est perpendiculaire à AB et DQ est perpendiculaire à BC, l’angle DRE et l’angle DQF sont tous deux des angles droits. c. On peut donc conclure que angle BNE = angle BMF = angle DRE = angle DQF. Sur la base des preuves ci-dessus, nous pouvons conclure que BEDF est un losange, car ses quatre côtés sont congrus et ses angles sont isométriques aux angles du losange ABCD.

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