Math Problem Statement

solve

Given a rectangle ABCD with diagonals intersecting at the center and forming an angle of 58°, determine the height x. The base (CD) is 18 cm.

Solution

The diagram shows a rectangle ABCDABCD with diagonals intersecting at the center, and one of the angles formed by the diagonals is 5858^\circ. The length of the base of the rectangle CDCD is 18cm18 \, \text{cm}, and we need to determine the height xx.

Approach:

  1. The diagonals of a rectangle bisect each other at right angles.
  2. Since diagonals in a rectangle are equal, triangles formed by the diagonals (like ABD\triangle ABD) are isosceles.
  3. The angle 5858^\circ is part of triangle ABD\triangle ABD, and we need to find the height xx.

Using the fact that the diagonals form isosceles triangles and applying trigonometry:

  • In ABD\triangle ABD, we know:
    • Angle at the center = 5858^\circ
    • Half of the diagonal BDBD = 182=9cm\frac{18}{2} = 9 \, \text{cm}.

Let me calculate xx using trigonometry.The height xx of the rectangle is approximately 14.4cm14.4 \, \text{cm}.

Would you like further explanation or have any questions?

Here are five related questions to deepen your understanding:

  1. What are the properties of diagonals in a rectangle?
  2. How would you calculate the diagonal of this rectangle?
  3. Can this problem be solved using the Pythagorean theorem instead of trigonometry?
  4. What would happen to the value of xx if the angle was increased or decreased?
  5. How would you approach finding xx if the angle given was not in the center?

Tip: In any right triangle, trigonometric ratios like sine, cosine, and tangent are very useful when you know one angle and one side.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Geometry
Properties of Rectangles

Formulas

Trigonometric Ratios: sin(θ), cos(θ), tan(θ)
Relationship between diagonals in a rectangle

Theorems

Properties of Diagonals in a Rectangle
Right Triangle Trigonometry

Suitable Grade Level

Grades 9-10