In the rectangle, the angle between the diagonals is 50 degrees. The length of the base is 17 cm. Find the length of the side x.
Answer (1)
To solve this problem, we need to find the length of side x in a rectangle where the angle between the diagonals is given as 50 degrees and the length of the base (one side) is 17 cm.
Step-by-step Solution:
Understanding the Rectangle and its Diagonals:
In a rectangle, the diagonals intersect each other at the center of the rectangle and create angles with each other.
The diagonals of a rectangle are equal in length.
The angle between these diagonals is 50 degrees in this specific scenario.
Using Trigonometry: Cosine Rule in the Triangle:
Consider the triangle formed by half of the rectangle, specifically one that includes the diagonals intersecting at the center. The angle between the diagonals is given as 50 degrees.
The Cosine Rule can be used in one of the resulting triangles to relate the sides and angles.
Apply Trigonometry: Relation through Half-Angle
Since diagonals of the rectangle split it into two congruent right triangles, each right triangle formed will have an angle of 2 5 ∘ (which is half of the angle between diagonals).
We can use the known side of the rectangle (which is the base) in relation to this bisected angle to find the other side using trigonometric identities.
Calculation
Given the known base A B = 17 cm, and using the angle cosine:
cos ( 2 5 ∘ ) = 17 x
Solving for x :
x = 17 × cos ( 2 5 ∘ )
You can use a calculator to compute this:
x ≈ 17 × 0.9063 (approximation of cos ( 2 5 ∘ ) )
x ≈ 15.4 cm
Therefore, the length of the side x is approximately 15.4 cm.
Understanding this involves applying knowledge of trigonometry, especially within the context of geometric shapes like rectangles. This approach shows how different geometric properties and trigonometric relations can be used to solve for unknowns in given figures.
