Math Problem Statement

ABCD is a rectangle. Find the length of each diagonal. AC = (3y)/4 and BD = 6y - 3.

Solution

To solve this problem, let's analyze the equations and the geometry of the rectangle:

Problem Breakdown:

  1. Given:

    • ABCD is a rectangle.
    • AC (one diagonal) is expressed as 3y4\frac{3y}{4}.
    • BD (the other diagonal) is expressed as 6y36y - 3.
  2. Key Property of a Rectangle:

    • In a rectangle, the diagonals are equal in length. Hence, AC=BDAC = BD.
  3. Set the equations equal: 3y4=6y3\frac{3y}{4} = 6y - 3

Solution:

Step 1: Eliminate the fraction by multiplying through by 4:

3y=4(6y3)3y = 4(6y - 3)

Step 2: Simplify:

3y=24y123y = 24y - 12

Step 3: Rearrange to isolate yy:

3y24y=123y - 24y = -12 21y=12-21y = -12

Step 4: Solve for yy:

y=1221=47y = \frac{-12}{-21} = \frac{4}{7}

Step 5: Calculate the lengths of AC and BD:

  • Substitute y=47y = \frac{4}{7} into the expressions for ACAC and BDBD:
    • AC=3y4=3474=1228=37AC = \frac{3y}{4} = \frac{3 \cdot \frac{4}{7}}{4} = \frac{12}{28} = \frac{3}{7}.
    • BD=6y3=6473=2473=247217=37BD = 6y - 3 = 6 \cdot \frac{4}{7} - 3 = \frac{24}{7} - 3 = \frac{24}{7} - \frac{21}{7} = \frac{3}{7}.

Thus, the length of each diagonal is 37\boxed{\frac{3}{7}}.


Would you like a detailed step-by-step explanation of any part? Let me know if you have questions or need clarification!

Related Questions:

  1. How can we use the property of diagonals to determine unknown variables in other quadrilaterals?
  2. If one diagonal's formula includes a square term, how would the solution process change?
  3. What are some real-life applications of rectangle diagonal properties?
  4. How would the problem change if ABCD were a square instead of a rectangle?
  5. Can you solve for yy if one diagonal was twice the other?

Tip: Always check the geometry properties of the shape when solving problems involving diagonals, sides, or angles.

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Math Problem Analysis

Mathematical Concepts

Algebra
Properties of Rectangles
Equations

Formulas

Diagonal lengths are equal in a rectangle

Theorems

Diagonal equality property of rectangles

Suitable Grade Level

Grades 7-10