Chapter Closure
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In a 45-45-90 triangle, the hypotenuse is sqrt(2) times the length of the legs.
In what type of right triangle is the hypotenuse twice the length of one of its legs?
When you know two angles in a triangle, you can calculate the third angle by using the Triangle Angle Sum Theorem.
Missing Angle: 90^(∘)
Missing Sides: 2 and 2sqrt(2)
Missing Angles: 30^(∘) and 60^(∘)
Missing Side: 4sqrt(3)
Missing Angle: 30^(∘)
Missing Sides: 3 and 3sqrt(3)
The given triangle has two 45^(∘) angles. Therefore, by the Triangle Angle Sum Theorem we know that the third angle, which we can label θ, must be a right angle.
θ + 45^(∘)+ 45^(∘)=180^(∘) ⇔ θ = 90^(∘)
Also, in a 45^(∘)-45^(∘)-90^(∘) triangle the hypotenuse is sqrt(2) times longer than any of its legs. With this information we can determine the hypotenuse. hypotenuse: legsqrt(2)= 2sqrt(2)
Examining the triangle, we see that it is a right triangle. We also see that the length of the hypotenuse is twice the length of one of the triangle's legs. This fits the description of a 30^(∘)-60^(∘)-90^(∘) triangle.
In a 30^(∘)-60^(∘)-90^(∘) triangle, if the shorter leg is a units then the second leg is sqrt(3)a units and the hypotenuse is 2a units. We can determine the length of the longer leg.
Again, this is a right triangle with a second angle of 60^(∘). By the Triangle Angle Sum Theorem we know that the third unknown angle is 30^(∘), which makes this a 30^(∘)-60^(∘)-90^(∘) triangle.
Like in Part B, if we have a 30^(∘)-60^(∘)-90^(∘) triangle the hypotenuse is always twice the length of the shorter leg, while the longer leg is sqrt(3) times the length of the shorter leg. Now we can identify the lengths of the remaining legs.
