a In a 30^(∘)-60^(∘)-90^(∘) triangle the shorter leg is half the length of the hypotenuse, and the longer leg is sqrt(3) times the length of the shorter leg. With this information, we can identify the missing side lengths.
Short leg:& 1 unit
Long leg:& 1 (sqrt(3))= sqrt(3) units
Hypotenuse:& 1(2)=2 units
Let's add the missing side lengths to the graph.
In this type of triangle the hypotenuse is always sqrt(2) times greater than its legs. With this information we can determine the length of the hypotenuse.
Legs:& 1 units
Hypotenuse:& 1(sqrt(2))=sqrt(2) units
Now we can complete the diagram.
b An equilateral triangle is a triangle with three congruent sides. From Part A, we know that the hypotenuse of a 30^(∘)-60^(∘)-90^(∘) triangle is always twice the length of the shorter leg. Therefore, if we put the longer legs of two 30^(∘)-60^(∘)-90^(∘) triangles up against each other, they will together form an equilateral triangle.
