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What is Drawing a Quadrilateral Similar to a Given Quadrilateral?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Drawing a quadrilateral similar to a given quadrilateral means creating a new quadrilateral that has the exact same shape but a different size. All corresponding angles remain equal, and all corresponding sides are in the same proportion (scaled up or down by the same factor).
Simple Example
Quick Example
Imagine you have a photo of your school building (which is a quadrilateral). If you print that photo in a smaller size for your album, or enlarge it to put on a poster, the smaller or larger photo of the building is similar to the original. The building's shape doesn't change, only its size.
Worked Example
Step-by-Step
Let's draw a quadrilateral ABCD with sides AB=4cm, BC=3cm, CD=5cm, DA=6cm, and diagonal AC=5cm. Now, let's draw a similar quadrilateral A'B'C'D' where each side is twice the original.
1. First, draw the original quadrilateral ABCD using a ruler and compass. Start by drawing base AB=4cm. Then, with A as center and radius 6cm, draw an arc for D. With B as center and radius 3cm, draw an arc for C. Use diagonal AC=5cm to locate C (from A) and D (from A) more accurately by intersecting arcs.
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2. To draw the similar quadrilateral A'B'C'D' with sides twice as long, calculate the new side lengths: A'B' = 2 * 4cm = 8cm, B'C' = 2 * 3cm = 6cm, C'D' = 2 * 5cm = 10cm, D'A' = 2 * 6cm = 12cm. The new diagonal A'C' will be 2 * 5cm = 10cm.
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3. Draw the new base A'B' = 8cm. This will be the base of your new quadrilateral.
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4. From point A', draw an arc with radius 12cm (for D'). From point B', draw an arc with radius 6cm (for C').
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5. Now, use the diagonal A'C'. From A', draw another arc with radius 10cm. The intersection of this arc with the arc from B' (radius 6cm) gives point C'.
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6. Connect B' to C'. From C', draw an arc with radius 10cm (for D'). The intersection of this arc with the arc from A' (radius 12cm) gives point D'.
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7. Connect A'D', B'C', C'D'. You now have quadrilateral A'B'C'D'.
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ANSWER: The quadrilateral A'B'C'D' is similar to ABCD, with each side twice as long. All angles remain the same.
Why It Matters
Understanding similarity is crucial in many fields. Architects use it to create scaled models of buildings, and engineers use it to design smaller prototypes before building large structures like bridges or machines. In computer graphics, it helps in resizing images or 3D models without distorting them.
Common Mistakes
MISTAKE: Changing the angles of the quadrilateral while scaling the sides. | CORRECTION: Remember that for similar shapes, only the side lengths change proportionally; all corresponding angles must remain exactly the same.
MISTAKE: Scaling different sides by different factors (e.g., making one side double, another triple). | CORRECTION: To maintain similarity, ALL corresponding sides must be scaled by the SAME factor. If one side is doubled, all sides must be doubled.
MISTAKE: Confusing similar quadrilaterals with congruent quadrilaterals. | CORRECTION: Congruent quadrilaterals are identical in both shape and size. Similar quadrilaterals have the same shape but can be different in size.
Practice Questions
Try It Yourself
QUESTION: A quadrilateral has sides 2cm, 4cm, 3cm, 5cm. If you draw a similar quadrilateral where the smallest side is 4cm, what will be the lengths of the other sides? | ANSWER: The scale factor is 4cm/2cm = 2. So, the other sides will be 4cm * 2 = 8cm, 3cm * 2 = 6cm, and 5cm * 2 = 10cm.
QUESTION: If Quadrilateral PQRS is similar to Quadrilateral WXYZ, and angle P = 70 degrees, angle Q = 110 degrees, angle R = 80 degrees, what are the measures of angle W, angle X, and angle Y? | ANSWER: Since they are similar, corresponding angles are equal. So, angle W = angle P = 70 degrees, angle X = angle Q = 110 degrees, and angle Y = angle R = 80 degrees.
QUESTION: A rectangular playground is 20m long and 15m wide. You want to make a scale model of it for a school project, using a scale factor of 1:100 (meaning 1 unit in the model represents 100 units in real life). What will be the length and width of your model playground in centimeters? | ANSWER: Length = 20m / 100 = 0.2m = 20cm. Width = 15m / 100 = 0.15m = 15cm. The model will be 20cm long and 15cm wide.
MCQ
Quick Quiz
Which of the following statements is TRUE for similar quadrilaterals?
Their areas are always equal.
Their corresponding angles are equal.
Their corresponding sides have different proportions.
They must be the exact same size.
The Correct Answer Is:
B
For similar quadrilaterals, the shape is preserved, meaning all corresponding angles are equal. Only the size changes, so areas are not always equal, and sides are proportional, not necessarily different proportions.
Real World Connection
In the Real World
When you use Google Maps or any navigation app on your phone, the map you see is a scaled-down, similar version of the actual roads and buildings. The map's shapes (like the quadrilaterals formed by road intersections) are similar to their real-world counterparts, allowing you to understand directions and distances proportionally.
Key Vocabulary
Key Terms
QUADRILATERAL: A four-sided polygon. | SIMILAR: Having the same shape but possibly different sizes. | SCALE FACTOR: The ratio by which all corresponding lengths in a shape are multiplied to create a similar shape. | CORRESPONDING SIDES: Sides that are in the same relative position in two similar shapes. | CORRESPONDING ANGLES: Angles that are in the same relative position in two similar shapes.
What's Next
What to Learn Next
Great job understanding similarity! Next, you can explore 'Area and Perimeter of Similar Quadrilaterals'. This will teach you how the area and perimeter change when you scale a quadrilateral, building directly on what you learned about side lengths.
