a The lengths of the leg and hypotenuse are both multiples of 4.
B
b The hypotenuse in a 45^(∘)-45^(∘)-90^(∘) triangle is the length of one of the triangle's legs multiplied by sqrt(2).
C
c The lengths of the leg and hypotenuse are both multiples of 2.
D
d What are the relative lengths of a 30^(∘)-60^(∘)-90^(∘) triangle's sides?
A
aUnknown Leg: 16 inches
B
bHypotenuse: 4sqrt(2) yards
Unknown Leg: 4 yards
C
cUnknown Leg: 24 feet
D
dShort Leg: 10 meters
Long Leg: 10sqrt(3) meters
Practice makes perfect
a If we scale down the triangle as far as we can and get either a 3-4-5 triangle or a 5-12-13 triangle, we know the triangle is a Pythagorean triple. In the given triangle the hypotenuse is 20 inches and one leg is 12 inches. These numbers are both multiples of 4, which means we can scale down the triangle by dividing all of its sides by 4.
Now we know that the given triangle is a dilated version of a 3-4-5 triangle where the unknown side in the given triangle corresponds to the leg in a 3-4-5 triangle with the length of 4. We can write and solve an equation for x.
x/4=4 ⇔ x=16 inches
In a 45^(∘)-45^(∘)-90^(∘) triangle the hypotenuse is always sqrt(2) times longer than the length of the legs. Now we can determine the length of the hypotenuse.
Hypotenuse: legsqrt(2)= 4sqrt(2)
c Like in Part A, if we scale down the triangle as far as we can and get either a 3-4-5 triangle or a 5-12-13 triangle, we know the triangle is a Pythagorean Triple.
In the given triangle the hypotenuse is 26 meters and one leg is 10 meters. These lengths are both multiples of 2, which means we can scale down the triangle by dividing all of the sides by 2.
The given triangle is a dilated version of a 5-12-13 triangle where the unknown side in the given triangle corresponds to the leg in a 5-12-13 triangle that has a length of 12 units. We can write and solve an equation for the unknown side.
x/2=12 ⇔ x=24 inches
d Since this is a right triangle with a second known angle of 60^(∘), we know that this is a 30^(∘)-60^(∘)-90^(∘) triangle — a Pythagorean Triple. In this type of a triangle if the shortest leg is a units then the longer leg is sqrt(3)a units and the hypotenuse is 2a units.
In the given triangle the hypotenuse is 20 meters. If we equate this measure with 2a we can find the measure of a, which allows us to calculate the length of the legs.
The short leg is 10 meters. Finally, we will multiply this measure by sqrt(3) to determine the length of the longer leg.
Longer leg: sqrt(3)( 10)=10sqrt(3)
