Question

Given: The coordinates of isosceles trapezoid JKLM are J(-b, c), K(b, c), L(a, 0), and M(-a, 0). Prove: The diagonals of an isosceles trapezoid are congruent. As part of the proof, find the length of km

Answer 1

We have to find the lengths of the diagonals KM and JL:
d ( KM ) = √ (( - a - b )² + ( 0 - c )²) = √ (( a + b )² + c² )
d ( JL ) = √ ( ( a - ( - b ) )² + ( 0 - c )²) = √ ( ( a + b )² + c² )
So the lengths of the diagonals KM and JL are congruent.
The lengths of the diagonals of the isosceles trapezoid are congruent.   

Answer 2

Answer:

The diagonals of an isosceles trapezoid are congruent and the length of KM is  $\sqrt{a^{2}+b^{2}+2ab +c^{2}}\ units$ .

Step-by-step explanation:

Formula

$Distance\ formula = \sqrt{(x_{2}-x_{1})^{2} +(y_{2}-y_{1})^{2} }$

As given

The coordinates of isosceles trapezoid JKLM are J(-b, c), K(b, c), L(a, 0), and M(-a, 0).

The diagram is shown below.

Now find out the length of the diagonal.

As the diagonal is JL .

The coordinates of the JL are J (-b,c) and L (a,o)

Putting the value in the above

$JL= \sqrt{(a-(-b))^{2} +(0-c)^{2} }$

$JL= \sqrt{(a+b)^{2} +(-c)^{2} }$

$JL= \sqrt{(a+b)^{2} +c^{2}}$

(As by using the formula(a + b)² = a² + b² +2ab )

Put this in the above

$JL= \sqrt{a^{2}+b^{2}+2ab +c^{2}}\ units$

Now find the length of diagonal KM .

As coordinates of K (b,c) and M (-a,0).

$KM = \sqrt{(-a-b)^{2} +(0-c)^{2}}$

$KM = \sqrt{(b+a)^{2} +c^{2}}$

(As by using the formula(a + b)² = a² + b² +2ab )

$KM= \sqrt{a^{2}+b^{2}+2ab +c^{2}}\ units$

As the length of the diagonal JL and KM are equal .

Thus the diagonals of an isosceles trapezoid are congruent and the length of KM is  $\sqrt{a^{2}+b^{2}+2ab +c^{2}}\ units$ .