Question
18.7 Chech Your Underntandions
is a triangle where
and
What is the length of
4
1 Brat
4
1 Brat
Ask by May Reyes. in the United States
Jan 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The length of
is
.
Solution
Let’s solve the problem step by step.
Given:
- Triangle
with a right angle at ( ). -
, which makes (since the sum of angles in a triangle is ). - The length of side
.
Objective:
- Find the length of
.
Solution:
In a
triangle, the sides are in the ratio
, where:
- The side opposite the
angle ( ) is the shortest side. - The side opposite the
angle ( ) is times the shortest side. - The hypotenuse (
) is twice the shortest side.
Given that
, we can set up the following relationship based on the ratio:
So, the length of
is
.
Answer:
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Mind Expander
To find the length of
in triangle
, we can apply trigonometric ratios or utilize the properties of the 30-60-90 triangle. Since
and
, we realize that angle
implies that
acts as the diagonal across from the 60-degree angle.
In a 30-60-90 triangle, the ratios of the lengths of the sides opposite the angles are
. Given that
, which is opposite the 60-degree angle, we can conclude that
or directly apply the ratio, giving us the leg opposite
.
Now, let’s do some fun math! The length of
can be calculated as
, which leads us to a magical number!
By calculating it out, we find
. So, it’s delightful to see that the answer is indeed
!
Adding a sprinkle of trivia, the 30-60-90 triangle is not just a key player in trigonometry; it appears in architecture and design. The proportions make it perfect for creating aesthetically pleasing triangular designs in various structures!
For those intrigued by more math wonders, I encourage exploring more about the various types of triangles! Diving into 45-45-90 triangles and their properties can reveal additional delightful relationships and real-world applications in everything from engineering to art. Happy calculating!