1. ## Rhombus

The sides of a rhombus are 10cm long. If the lengths of the diagonals differ by 4, what is the area of the rhombus?

2. Hello, jumpman23!

The sides of a rhombus are 10cm long.
If the lengths of the diagonals differ by 4, what is the area of the rhombus?

You're expected to know a few facts about a rhombus.

. . The diagonals of a rhombus are perpendicular.

. . If the diagonals are $\displaystyle d_1$ and $\displaystyle d_2$, the area is: .$\displaystyle A \:=\:\frac{1}{2}d_1d_2$
Code:
            A     10    B
* - - - - - *
/ *       * /
/   *   *   /
10/     *O    / 10
/   *   *   /
/ *       * /
* - - - - - *
D    10     C

We have: .$\displaystyle AB \,=\, BC \,=\, BD \,=\, DA \,=\, 10$

Let $\displaystyle AC\,= \,d$. .Then $\displaystyle BD\,=\,d + 4$

Then: .$\displaystyle AO \,= \,\frac{d}{2},\;BO \,= \,\frac{d+4}{2}$

In right triangle $\displaystyle AOB :\;\;AO^2 + BO^2\:=\:AB^2$

. . so we have: .$\displaystyle \left(\frac{d}{2}\right)^2 + \left(\frac{d+4}{2}\right)^2\;=\;10^2$

. . which simplifies to: .$\displaystyle d^2 + 4d - 192\:=\:0$

. . which factors: .$\displaystyle (d -12)(d + 16)\:=\:0$

. . and has roots: .$\displaystyle d \:=\:12,\,-16$

The diagonals are: .$\displaystyle AC \,=\, d \,= \,12,\;BD \,= \,d+4 \,= \,16$

Therefore, the area is: .$\displaystyle A \:=\:\frac{1}{2}(12)(16)\:=\:96$ cm².