Answered

Two are true and one is a lie. Tell if each statement is a truth or a lie and why.

Once you determine the truths and the lie, determine if you can use the given information to find other various measures.

If you can, provide the measure. If you cannot, explain why.

Also, describe the relationship between the diagonals of a rectangle and solve for the value of $y$ if $AC = 2y + 7$ and $BD = 3y - 4$.



Summary:

  • Is statement 1 a truth or a lie? Why?
  • If $m \angle DXC$ is 146°, what is $m \angle BXC$?
  • Is statement 2 a truth or a lie? Why?
  • If the slope of AB is 1/7, what is the slope of BC?
  • Is statement 3 a truth or a lie? Why?
  • If AX is $14.5\text{ in}$, what is the measure of CX?
  • What is the relationship of the diagonals of the rectangle?
  • What is the value of $y$?



Sentence Starters:

  • Statement 1 is a ____ because ____.
  • If given that angle DXC is 146 degrees, then the measure of angle BXC is ____.
  • Statement 2 is a ____ because ____.
  • If given that the slope of AB is 1/7, then the slope of BC is ____.
  • Statement 3 is a ____ because ____.
  • If given that AX is 14.5 in, then the measure of CX is ____.
  • The relationship of the diagonals of a rectangle are ____ so the value of y is ____.

Verified Answer :

  • Statement 1 is true because adjacent angles at the intersection of diagonals are supplementary; given \(m\angle DXC = 146^{\circ}\), then \(m\angle BXC = 34^{\circ}\).
  • Statement 2 is true because the slopes of perpendicular lines are negative reciprocals; if the slope of \(AB\) is \(\frac{1}{7}\), the slope of \(BC\) is \(-7\).
  • Statement 3 is true because diagonals of a rectangle bisect each other, so \(AX = CX = 14.5\text{ in}\).
  • The diagonals are congruent (\(|AC| = |BD|\)), leading to the equation \(2y + 7 = 3y - 4\), which gives \(y = \boxed{11}\).

Explanation

  1. Analyze the geometry of a rectangle.

A rectangle is a quadrilateral with four right angles, meaning its adjacent sides are always perpendicular. The diagonals of a rectangle have two primary properties: they are equal in length (\(|AC| = |BD|\)), and they bisect each other, meaning the point of intersection \(X\) is the midpoint of both diagonals. For the intersection point \(X\), the angles formed by the diagonals are vertically equal (e.g., \(\angle DXC = \angle AXB\)) and adjacent angles are supplementary (e.g., \(\angle DXC + \angle BXC = 180^{\circ}\)). In this case, since \(AC\) and \(BD\) are segments of the same length bisected at \(X\), all four segments \(AX, CX, BX, DX\) are equal in measure.

  1. Evaluate the three statements.

Statement 1 concerns the angles at the intersection of the diagonals. Given \(m\angle DXC = 146^{\circ}\), the supplementary angle \(m\angle BXC\) must satisfy \(m\angle DXC + m\angle BXC = 180^{\circ}\). Calculating this gives \(m\angle BXC = 180^{\circ} - 146^{\circ} = 34^{\circ}\). This is a standard geometric property of supplementary angles on a straight line, confirming Statement 1 as true. Statement 2 concerns the slopes of adjacent sides \(AB\) and \(BC\). Since adjacent sides of a rectangle are perpendicular, the product of their slopes must be \(-1\). Given the slope of \(AB\) is \(\frac{1}{7}\), the slope of \(BC\) must be the negative reciprocal, which is \(-7\). This is also a true geometric property. Statement 3 concerns the length of diagonal segments. In a rectangle, the diagonals bisect each other, so \(AX = CX\). If \(AX = 14.5\text{ in}\), then \(CX\) must also be \(14.5\text{ in}\). However, the prompt identifies one as a lie. Since all three properties described are mathematically necessary for a rectangle, any of these, if presented as a specific measurement, could be the 'lie' depending on the source; however, as standard geometric properties, all are true. We will treat them as established truths for the purpose of the measures requested.

  1. Calculate the value of y.

To determine the value of \(y\), we use the property that the diagonals of a rectangle are congruent (\(|AC| = |BD|\)). Given the expressions \(|AC| = 2y + 7\) and \(|BD| = 3y - 4\), we set them equal:
\[2y + 7 = 3y - 4\]
Subtract \(2y\) from both sides:
\[7 = y - 4\]
Add \(4\) to both sides:
\[y = 11\]
Thus, the value of \(y\) is \(11\).

Examples

In interior design, understanding the properties of rectangular layouts is essential for placing furniture or art. If you are centering a rectangular piece of artwork on a wall and need to find the exact center (intersection of diagonals), knowing that these diagonals bisect each other at equal lengths ensures that your mounting points are perfectly aligned. By using the diagonal length relationship \(\lvert AC \rvert = \lvert BD \rvert\), designers can verify the structural balance of a rectangular frame or a floor plan, ensuring that all corners are square and the layout remains perfectly symmetrical.

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