Question

these figures are congruent. what is de? centimeters

Explanation:

Step1: Recall Congruent Triangles Property

Congruent triangles have corresponding sides equal. So, we need to find the corresponding side of \( DE \) in the other triangle.

Step2: Identify Corresponding Sides

In triangle \( ECD \) (right - angled at \( C \)) and triangle \( NPO \), since the triangles are congruent, we look at the sides. The side \( NP = 58\)? Wait, no, in triangle \( ECD \), angle \( E = 31^{\circ}\), angle \( D=59^{\circ}\), right - angled at \( C \). In triangle \( NPO \), angle \( N = 31^{\circ}\), angle \( P = 59^{\circ}\). The side \( NP = 58\)? Wait, no, let's match the corresponding sides. The side \( EC = 50\) cm, \( NC\)? Wait, no, the triangle \( ECD \) has sides \( EC = 50\) cm, \( CD = 30\) cm, and right - angled at \( C \). The other triangle \( NPO \) has sides \( NP = 58\)? Wait, no, wait the length of \( NP \) is 58 cm? Wait, no, let's check the angles. In triangle \( ECD \), angles are \( 31^{\circ}\), \( 59^{\circ}\), \( 90^{\circ}\). In triangle \( NPO \), angles are \( 31^{\circ}\), \( 59^{\circ}\), so the third angle in \( NPO \) is \( 180-(31 + 59)=90^{\circ}\)? Wait, no, the triangle \( NPO \) is not right - angled? Wait, no, the problem says the figures are congruent. So triangle \( ECD\) (right - angled at \( C\)) is congruent to triangle \( NPO\). So the corresponding sides: \( EC\) corresponds to \( NO\) (50 cm), \( CD\) corresponds to \( PO\), and \( DE\) corresponds to \( NP\). Wait, in triangle \( NPO\), the length of \( NP\) is 58 cm? Wait, the side \( NP\) is 58 cm? Let's check the triangle \( ECD\): using Pythagoras, \( DE=\sqrt{EC^{2}+CD^{2}}=\sqrt{50^{2}+30^{2}}=\sqrt{2500 + 900}=\sqrt{3400}\)? No, that's not right. Wait, no, maybe I made a mistake. Wait, the triangle \( NPO\) has side \( NP = 58\) cm? Wait, the side \( NP\) is 58 cm. Since the triangles are congruent, \( DE\) (the hypotenuse of the right - angled triangle \( ECD\)) should be equal to \( NP\) (the side of the other triangle). Wait, in triangle \( NPO\), the length of \( NP\) is 58 cm. So \( DE=NP = 58\) cm? Wait, let's see: in triangle \( ECD\), angle \( E = 31^{\circ}\), angle \( D = 59^{\circ}\), right - angled at \( C\). In triangle \( NPO\), angle \( N=31^{\circ}\), angle \( P = 59^{\circ}\), so the sides: \( NO = 50\) cm (corresponding to \( EC = 50\) cm), \( PO\) (corresponding to \( CD = 30\) cm), and \( NP\) (corresponding to \( DE\)) has length 58 cm. Alternatively, in triangle \( ECD\), using the Law of Sines: \(\frac{DE}{\sin90^{\circ}}=\frac{EC}{\sin59^{\circ}}=\frac{CD}{\sin31^{\circ}}\). But since the triangles are congruent, the length of \( DE\) should be equal to the length of \( NP\), which is 58 cm. Wait, the side \( NP\) is 58 cm (as given in the diagram: \( NP = 58\) cm). So since the triangles are congruent, \( DE=NP = 58\) cm.

Answer:

58