3-sided
How many triangles can Jack draw with two sides of lengths $6$cm and $8$cm and an area of $7$cm$^2$?
Age
14 to 16
Challenge level
Problem
Jack's teacher asked him to draw a triangle of area $7\textrm{ cm}^2$.
Two sides are to be of length $6\textrm{ cm}$ and $8\textrm{ cm}$.
How many possibilities are there for the length of the third side of the triangle?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Let the base $AB$ of the triangle be the side of length $8\textrm{ cm}$ and let $AC$ be the side of length $6\textrm{ cm}$. So $C$ must lie on the circle with centre $A$ and radius $6\textrm{ cm}$ as shown. The area of the triangle is to be $7\textrm{ cm}^2$, so the perpendicular from $C$ to $AB$ (or to $BA$ produced) must be of length $\frac{7}{4}\textrm{cm}$.
Image
The diagram shows the four possible positions $D$, $E$, $F$ and $G$ of $C$. However, since $∠BAG = ∠BAF$ and $∠BAD = ∠BAE$, these correspond to exactly two possibilities for the length of the third side $AC$. The diagrams below show the two possibilities.
Image
