Let's consider an equilateral triangle with sides of length . Notice that any altitude will split it into two congruent triangles, as shown below.
In the new smaller triangle the acute angles will be and Such triangles are called -- triangles.
Based on the picture above, we see a special property shared by all -- triangles:
In any -- triangle, the hypotenuse is twice as long as the shortest leg (which is opposite the angle).
This also works in the other direction:
If the hypotenuse is twice as long as the shortest leg, then the given right triangle is a -- triangle.
What is the length of the hypotenuse in a -- triangle if the side opposite the angle has a length of
Let's refer to our standard -- triangle.
In any -- triangle, the hypotenuse is twice the length of the shorter leg.
The leg opposite the angle is the shorter leg of the triangle. So here, the shorter leg has length
So here, the length of the hypotenuse must be
What is the length of the shortest leg in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the hypotenuse has a length of $14\,\textrm{m}?$
|
a
|
$5\sqrt 2\,\textrm{m}$ |
|
b
|
$\dfrac{7}{\sqrt 2}\,\textrm{m}$ |
|
c
|
$7\sqrt 2\,\textrm{m}$ |
|
d
|
$2\sqrt 7\,\textrm{m}$ |
|
e
|
$7\,\textrm{m}$ |
What is the length of the hypotenuse in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the side opposite the $30^\circ$ angle has a length of $8?$
|
a
|
$\dfrac 1 4$ |
|
b
|
$8$ |
|
c
|
$\sqrt{8}$ |
|
d
|
$16$ |
|
e
|
$4$ |
Consider the same -- triangle from before. This time, however, we will focus on the height () instead of the hypotenuse.
Applying the Pythagorean Theorem, we have
which gives us the second important property:
In any -- triangle, the length of the longer leg (which is opposite the angle) is times the length of the shorter leg (which is opposite the angle).
We can summarize everything we know about -- triangles using the diagram below:
In the triangle below, what is the length of the shorter leg?
Let's refer to our standard -- triangle.
Since this is a -- triangle, the length of the longer leg is times that of the shorter leg.
Here, the longer leg has length and the shorter leg has length so we have
To remove the radical from the denominator, we can multiply the numerator and denominator by the radical:
So, the length of the shorter leg is
In the triangle above, what is the length of the shorter leg?
|
a
|
$2\sqrt{3}$ |
|
b
|
$\sqrt{3}$ |
|
c
|
$3$ |
|
d
|
$1$ |
|
e
|
$2$ |
In the triangle above, what is the length of $\overline{AC}?$
|
a
|
$6\sqrt 3$ |
|
b
|
$6\sqrt 2$ |
|
c
|
$12\sqrt{3}$ |
|
d
|
$12$ |
|
e
|
$3\sqrt{3}$ |
What is the length of the longer leg in a -- triangle if the hypotenuse has a length of
Let's refer to our standard -- triangle.
Since the hypotenuse has a length of we have
Now that we know we can calculate the length of the longer leg as follows:
Therefore, the longer leg has a length of
What is the length of the hypotenuse in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the longer leg has a length of $3?$
|
a
|
$6$ |
|
b
|
$2\sqrt 2$ |
|
c
|
$2\sqrt 3$ |
|
d
|
$3\sqrt 3$ |
|
e
|
$3\sqrt 2$ |
What is the length of the longer leg in a $30^\circ$-$60^\circ$-$90^\circ$ triangle if the hypotenuse has a length of $10?$
|
a
|
$\dfrac 1 2$ |
|
b
|
$\sqrt{3}$ |
|
c
|
$\dfrac {\sqrt{3}} 2$ |
|
d
|
$5$ |
|
e
|
$5\sqrt{3}$ |
