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Chapter 1: Problem 10
Sketch and label the figure. Mark the figures. Rectangle \(R G H T\) with diagonals \(\overline{R H}\) and \(\overline{G T}\)intersecting at \(I\)
Short Answer
Expert verified
Draw a rectangle labeled as R, G, H, T with diagonals \(\rightharpoonup{RH}\) and \(\rightharpoonup{GT}\) intersecting at point I.
Step by step solution
01
- Draw the Rectangle
Sketch a rectangle and label its vertices as R, G, H, and T in a clockwise or counterclockwise manner.
02
- Draw the Diagonals
Draw diagonal \(\rightharpoonup{RH}\) from vertex R to vertex H, and diagonal \(\rightharpoonup{GT}\) from vertex G to vertex T. Make sure they intersect inside the rectangle.
03
- Mark the Intersection
Label the point of intersection of the diagonals (\(\rightharpoonup{RH}\) and \(\rightharpoonup{GT}\)) as point I.
04
- Label the Figure
Write down the labels of vertices (R, G, H, T) and point I at the correct positions on the rectangle.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rectangle Properties
A rectangle is a four-sided polygon where every interior angle is a right angle (90 degrees). This means all opposite sides are equal in length. Since opposite sides are parallel, you will often encounter properties like:
- The perimeter of a rectangle is calculated as: \ P = 2(l + w) \ where \( l \) is the length and \( w \) is the width.
- The area of a rectangle is given by \ A = l \times w.
The key feature of a rectangle is its symmetry about its diagonals. Let’s explore this in the next section.
Diagonals Intersection
In a rectangle, the diagonals are important lines that connect opposite corners. A remarkable property of the diagonals is that they intersect each other at the center of the rectangle, and they are of equal length. Mathematically, this means:
- The diagonal lengths are given by \ d = \sqrt{l^2 + w^2}
- They bisect each other, meaning each half of the diagonal is equal.
For the rectangle \( RGHT \), the diagonals \( \overline{RH} \) and \( \overline{GT} \), intersect at point \( I \). This point is equidistant from all four corners.
Labeling Points
Correct labeling is crucial for understanding geometry figures. When you label a rectangle, start by naming the vertices. Typical labels might be clockwise (\( R, G, H, T \)) or counterclockwise. Always ensure the points follow a logical order. After labeling the vertices, you can then draw and label the diagonals, making sure the point of intersection is marked clearly. In our case, point \( I \) is where the diagonals \( \overline{RH} \) and \( \overline{GT} \) meet. Use clear and distinct labels to avoid confusion.
Geometry Sketching
Sketching geometrical figures helps in visualizing and understanding their properties. Here’s how to sketch correctly:
1. Start by accurately drawing the shape. For a rectangle, use a ruler to ensure all sides are straight.
2. Label the vertices as mentioned.
- Draw the diagonals from corner to corner.
- Mark the intersection point and label it clearly.
Refer to your labels multiple times to make sure they're correctly placed. Drawing accurately will help you calculate areas, perimeters, and other properties without errors.
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