Question

Please solve q 17?

Answer 1

`BC = 3sqrt 3`

Explanation:

We assume the quadrilateral has its vertices in alphabetical order (i.e. it can be called `ABCD`).

Since the diagonals of `ABCD` are perpendicular, their intersection creates 4 right triangles inside `ABCD`, and each side of the quadrilateral is the hypotenuse of one of these triangles.

Label the interior sides as `a, b, c, d` as shown below:

By the Pythagorean theorem, we get the following 4 equations:

(1) `a^2+b^2color(white)("  "+c^2+d^2)=4^2`
(2) `color(white)(a^2+)b^2+c^2color(white)("  "+d^2)=(BC)^2`
(3) `color(white)(a^2+b^2+)c^2+d^2=6^2`
(4) `a^2color(white)("  "+b^2+c^2)+d^2=5^2`

Adding equations (1) + (3) gives

(5) `a^2+b^2+c^2+d^2 = 4^2 + 6^2`

while adding (2) + (4) gives

(6) `a^2+b^2+c^2+d^2 = (BC)^2 + 5^2`

The left-hand sides of (5) and (6) are equal. This means their right hand sides must also be equal:

(7) `4^2+6^2 = (BC)^2 + 5^2`

Equation (7) shows us something very useful: in a convex quadrilateral with perpendicular diagonals, the two pairs of squared opposite sides have equal sums.

We now have an equation where `BC` is the only unknown. We can now solve for `BC`.

`16+36 = (BC)^2 + 25`

`=> (BC)^2 = 16+36 - 25`
`color(white)(=>(BC)^2)=27`

`=>BC = sqrt(27)`
`color(white)(=>BC)=3sqrt(3)`