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Chapter 8: Problem 5

In rhombus \(M N P Q,\) how does the length of the altitude from \(Q\) to \(\overline{P N}\) compare to the length of the altitude from \(Q\) to \(\overline{M N} ?\) Explain.

Short Answer

Expert verified
The altitudes are equal in length.

Step by step solution

01

Understand the Problem

In a rhombus, all sides are equal in length and opposite angles are equal. We need to compare the length of two altitudes drawn from point \( Q \) to lines \( \overline{PN} \) and \( \overline{MN} \).
02

Draw the Altitudes

Draw two lines from point \( Q \) that are perpendicular to the sides \( \overline{PN} \) and \( \overline{MN} \). The lengths of these perpendiculars are the altitudes we need to compare.
03

Property of Rhombus

In a rhombus, opposite sides are parallel, meaning \( \overline{PN} \parallel \overline{QM} \) and \( \overline{MN} \parallel \overline{QP} \). The distances between parallel lines in a rhombus are equal. As a result, the length of the altitude from \( Q \) to \( \overline{MN} \) is the same as the length of the altitude from \( Q \) to \( \overline{PN} \).
04

Conclusion

Both altitudes from \( Q \) are equal in length because they measure the perpendicular distance between parallel opposite sides \( \overline{MN} \) and \( \overline{PN} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Altitude in Rhombus
In geometry, when discussing a rhombus, the term "altitude" typically refers to the perpendicular distance from a vertex to the opposite side, often referred to as the "base" for that particular altitude. In a rhombus, due to its symmetrical properties, altitudes play a vital role in calculating various geometric features such as the area.
  • The altitude in a rhombus is a key component in determining the height of the shape, relative to any of its sides.
  • Each side of a rhombus can act as a base, creating different perspectives on measuring the perpendicular height.
  • Concisely, an altitude is the shortest distance from a vertex to the line containing the opposite side.
To find the altitude, one would typically drop a perpendicular line from a vertex to the base line. This line ensures that the angle between the line and the base is 90 degrees (a right angle), which is essential for accurate measurement.
Perpendicular Distance
Understanding perpendicular distance in the context of geometric shapes like a rhombus is fundamental. It refers to the shortest line segment that can be drawn from a point to a line or between two parallel lines.
  • In a rhombus, perpendicular distances are crucial when calculating altitudes.
  • These lines are always at a right angle (90 degrees) to the line it's drawn to, ensuring it's the shortest possible distance.
  • The concept of perpendicular distance helps identify precise measurements and supports properties like symmetry.
In our specific exercise, to compare altitudes, each altitude from point \( Q \) to lines \( \overline{PN} \) and \( \overline{MN} \) acts as these perpendicular distances. Understanding this ensures we appreciate how altitude provides the shortest path between a vertex and the 'base', reinforcing the consistency across different altitudinal perspectives.
Parallel Sides in Geometry
Rhombuses are characterized by their parallel sides, a fundamental aspect when solving geometric problems involving these shapes. Each pair of opposite sides in a rhombus is parallel, simplifying the understanding and calculation of distances like altitudes.
  • Parallel sides in a rhombus point out how each pair remains equidistant.
  • This consistent distance helps to deduce that the length of altitudes measured perpendicular to these parallel lines will be the same.
  • Knowing the properties of parallelism assures that any perpendicular distance between these sides is constant and equal.
Thanks to these parallel properties, you can understand why altitudes such as the ones from \( Q \) to \( \overline{PN} \) and \( \overline{MN} \) are equal. The parallel nature maintains consistency in perpendicular measurements, which is critical in geometry when establishing equality between differing calculated altitudes.

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Most popular questions from this chapter

A trapezoid has an area of \(96 \mathrm{cm}^{2} .\) If the altitude has a length of \(8 \mathrm{cm}\) and one base has a length of \(9 \mathrm{cm},\) find the length of the other base.

Suppose that a circle of radius \(r\) is inscribed in a rhombus each of whose sides has length \(s .\) Find an expression for the area of the rhombus in terms of \(r\) and \(s\)

Use the formula \(A=\frac{1}{2} a P\) to find the area of the regular polygon described. Find the area of a regular pentagon with an apothem of length \(a=6.5\) in. and each side of length \(s=9.4\) in.

In a triangle whose area is 72 in \(^{2}\), the base has a length of 8 in. Find the length of the corresponding altitude.

The legs of an isosceles triangle each measure \(10 \mathrm{cm} .\) What are the limits of the length of the base?
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