Question

lamar is writing a coordinate proof to show that a segment from the midpoint of the hypotenuse of a right triangle to the opposite vertex forms two triangles with equal areas. he starts by assigning coordinates as given. 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, square) ). 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 ( square ). to find the area of ( \triangle mnl ), the length of the base, ( overline{ml} ), is ( square ), and the height is ( square ). 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 coordinates of \(N\)

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

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 \(a\), \(A=\frac{1}{2}\times2b\times a=ab\).

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

The length of \(ML\) is \(2a\) (since \(M(0,0)\) and \(L(2a,0)\)), and the height is \(b\) (the \(y\) - coordinate of \(N\)). Using the area formula \(A=\frac{1}{2}\times base\times height=\frac{1}{2}\times2a\times b = ab\).

Answer:

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