Question

enter the answers in the boxes to complete the coordinate proof
n is the midpoint of ( overline{kl} ). therefore, the coordinates of n are ( (a, quad) ).
to find the area of ( \triangle knm ), the length of the base, ( overline{mk} ), is ( 2b ), and the height is a. so an
expression for the area of ( \triangle knm ) is
to find the area of ( \triangle mnl ), the length of the base, ( overline{ml} ), is , and the height is . so an
expression for the area of ( \triangle mnl ) is ab.
comparing the expressions for the areas shows that the areas of the triangles are equal

Explanation:

Step1: Find the y - coordinate of the mid - point \(N\)

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). If \(K=(0,0)\) and \(L=(2a,0)\), for the mid - point \(N\) of \(\overline{KL}\), \(x=\frac{0 + 2a}{2}=a\) and \(y=\frac{0+0}{2}=0\).

Step2: Calculate the area of \(\triangle KNM\)

The area formula for a triangle is \(A=\frac{1}{2}\times base\times height\). Given base \(MK = 2b\) and height \(a\), then \(A=\frac{1}{2}\times2b\times a=ab\).

Step3: Determine the base and height of \(\triangle MNL\)

Since the area of \(\triangle MNL\) is \(A = ab=\frac{1}{2}\times base\times height\). Let the base be \(2b\) (because \(MK = 2b\) and \(MK=NL\) as \(N\) is the mid - point of \(KL\)) and height \(a\) (the vertical distance from \(M\) to \(KL\)).

Answer:

The coordinates of \(N\) are \((a,0)\). The expression for the area of \(\triangle KNM\) is \(ab\). The length of the base \(\overline{ML}\) is \(2b\) and the height is \(a\).