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

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

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` .

Answer 2

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.