Math Problem Statement

Quadrilateral RSTU is a rectangle with diagonal RT. If m∠RTU = 8x + 4 and m∠STR = 3x - 2, find the value of x.

Solution

We are tasked with solving for xx in the context of a rectangle RSTURSTU with diagonal RTRT, and angles given as mRTU=8x+4m\angle RTU = 8x + 4 and mSTR=3x2m\angle STR = 3x - 2.

Explanation:

  1. In a rectangle, the diagonals bisect each other and form two congruent triangles.
  2. Specifically, the angles mRTUm\angle RTU and mSTRm\angle STR are complementary because the sum of angles in a triangle is 180180^\circ, and mRST=90m\angle RST = 90^\circ in the rectangle.

Thus, the equation is: mRTU+mSTR=90m\angle RTU + m\angle STR = 90^\circ

Solve for xx:

Substitute the given expressions: (8x+4)+(3x2)=90(8x + 4) + (3x - 2) = 90

Combine like terms: 11x+2=9011x + 2 = 90

Subtract 2 from both sides: 11x=8811x = 88

Divide by 11: x=8x = 8

Final Answer:

x=8x = 8

Would you like further details or clarifications?


Here are five related questions to expand your understanding:

  1. What is the measure of RTU\angle RTU and STR\angle STR when x=8x = 8?
  2. How does the property of diagonals in a rectangle contribute to this solution?
  3. What other types of polygons have complementary angles in specific cases?
  4. How can you verify the solution geometrically?
  5. Can you apply this approach to parallelograms with different properties?

Tip: Always confirm angle relationships (like complementary or supplementary) in geometric problems to set up accurate equations.

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Properties of Rectangles
Complementary Angles

Formulas

m∠RTU + m∠STR = 90°

Theorems

The diagonals of a rectangle form congruent triangles
The sum of angles in a triangle is 180°

Suitable Grade Level

Grades 8-10