B) Solve ANY FOUR of the following.
1) Fill in the blank and rewrite the statement.
In a right angled triangle if two acute angles are 30 and 60° then the side opposite to 30° is
...... And side opposite to 60° is............ times the hypotenuse.
2) In APQR, LP = 90°, s is the midpoint of side QR = 10 cm what is the length of seg PS.
3) In AABC, if LB = LC = 45° then which of the following is the largest side?
4) In AXYZ, LX = 90°, LY = 60° if XZ = 573 cm what is the length of seg YZ.
Answers
feature_triangles
Acute, obtuse, isosceles, equilateral…When it comes to triangles, there are many different varieties, but only a choice few that are "special." These special triangles have sides and angles which are consistent and predictable and can be used to shortcut your way through your geometry or trigonometry problems. And a 30-60-90 triangle—pronounced "thirty sixty ninety"—happens to be a very special type of triangle indeed.
In this guide, we'll walk you through what a 30-60-90 triangle is, why it works, and when (and how) to use your knowledge of it. So let's get to it!
What Is a 30-60-90 Triangle?
A 30-60-90 triangle is a special right triangle (a right triangle being any triangle that contains a 90 degree angle) that always has degree angles of 30 degrees, 60 degrees, and 90 degrees. Because it is a special triangle, it also has side length values which are always in a consistent relationship with one another.
The basic 30-60-90 triangle ratio is:
Side opposite the 30° angle: x
Side opposite the 60° angle: x*√3
Side opposite the 90° angle: 2x
body_306090 traditional
For example, a 30-60-90 degree triangle could have side lengths of:
2, 2√3, 4
body_Example 1
7, 7√3, 14
body_example 2
√3, 3, 2√3
body_example_reverse.png
(Why is the longer leg 3? In this triangle, the shortest leg (x) is √3, so for the longer leg, x√3=√3*√3=√9=3. And the hypotenuse is 2 times the shortest leg, or 2√3)
And so on.
The side opposite the 30° angle is always the smallest, because 30 degrees is the smallest angle. The side opposite the 60° angle will be the middle length, because 60 degrees is the mid-sized degree angle in this triangle. And, finally, the side opposite the 90° angle will always be the largest side (the hypotenuse) because 90 degrees is the largest angle.
Though it may look similar to other types of right triangles, the reason a 30-60-90 triangle is so special is that you only need three pieces of information in order to find every other measurement. So long as you know the value of two angle measures and one side length (doesn't matter which side), you know everything you need to know about your triangle.
For example, we can use the 30-60-90 triangle formula to fill in all the remaining information blanks of the triangles below.
Example 1
body_demo 2
We can see that this is a right triangle in which the hypotenuse is twice the length of one of the legs. This means this must be a 30-60-90 triangle and the smaller given side is opposite the 30°.
The longer leg must, therefore, be opposite the 60° angle and measure 6*√3, or 6√3.
Example 2
body_demo 4
We can see that this must be a 30-60-90 triangle because we can see that this is a right triangle with one given measurement, 30°. The unmarked angle must then be 60°.
Since 18 is the measure opposite the 60° angle, it must be equal to x√3. The shortest leg must then measure
18
√3
.
(Note that the leg length will actually be
18
√3
*
√3
√3
=
18√3
3
=6√3 because a denominator cannot contain a radical/square root).
And the hypotenuse will be 2(
18
√3
)
(Note that, again, you cannot have a radical in the denominator, so the final answer will really be 2 times the leg length of 6√3 => 12√3).
Example 3
body_demo 3
Again, we are given two angle measurements (90° and 60°), so the third measure will be 30°. Because this is a 30-60-90 triangle and the hypotenuse is 30, the shortest leg will equal 15 and the longer leg will equal 15√3.
body_eight ball
No need to consult the magic eight ball—these rules always work.
Why It Works: 30-60-90 Triangle Theorem Proof
But why does this special triangle work the way it does? How do we know these rules are legit? Let's walk through exactly how the 30-60-90 triangle theorem works and prove why these side lengths will always be consistent.
First, let's forget about right triangles for a second and look at an equilateral triangle.
body_proof 1
An equilateral triangle is a triangle that has all equal sides and all equal angles. Because a triangle's interior angles always add up to 180° and
180
3
=60, an equilateral triangle will always have three 60° angles.
body_proof 2
Now let's drop down a height from the topmost angle to the base of the triangle.
body_proof 3
We've now created two right angles and two congruent (equal) triangles.
How do we know they're equal triangles? Because we dropped a height from an equilateral triangle, we've split the base exactly in half. The new triangles also share one side length (the height), and they each have the same hypotenuse length. Because they share three side lengths in common (SSS), this means the triangles are congruent.
body_proof 4
Note: not only are the two triangles congruent based on the principles of side-side-side lengths, or SSS, but also based on side-angle-side measures (SAS), angle-angle-side (AAS), and angle-side-angle (ASA). Basically? They're most definitely congruent .