(Length and Area Ratio s)

(Length and Area Ratio s) Standard Televisions are measured by the length of the diagonal. Most manufactures included the TV frame as part of the measurement (when measuring CRT (cathode ray tube) screens). So, you may want to find the diagonal measure of the viewable area which most manufactures provide in a much smaller print. When measuring Flat Panel LCD (liquid crystal display) screens, the manufactures just use the viewable area and do include the frame. In either type of screen, standard screens sides are always in a ratio of 4:3. This ratio just means that the height of the screen is exactly ¾ the length at the bottom and that all standard TV screens are similar figures. The latest portable DVD players usually have a 7 inch viewable screen area. Using Sketchpad, create a rectangle that has a 4:3 (length: height) ratio and 7 for the diagonal (Keep your measurements in cm. This will help keep the scale of things more manageable. ) 7 in. o Start creating a horizontal line segment at the bottom of the sketch. Double click on the left endpoint (point A) until it flashes (this denotes that the point has been marked as a center). o Click in a blank space to deselect everything. Highlight the right endpoint (point B). Select Dilate under the Transform menu. Change the ratio to ¾ and press Dilate. o Click in a blank space to deselect everything. Highlight the left endpoint (point A) and the dilated point (point C) in that order. Select Circle By Center+Point under the Construct Menu. o Click in a blank space to deselect everything. Highlight the left endpoint (point A) and the actual segment AB. Select Perpendicular Line under the Construct menu. o Using the selection tool, click on the precise top intersection of the circle and the perpendicular line. This should create a point of intersection. M. Winking p.28

o Click in a blank space to deselect everything. Highlight the circle, the perpendicular line, and the dilated point (point C). Make sure that the intersection point (point D) is NOT highlighted. Select Hide Objects under the Display menu. D o Using the segment tool construct a segment between the left endpoint (point A) and the intersection point (point D). o Click in a blank space to deselect everything. Select intersection point (point D) and the new segment (segment AD). Select Perpendicular Line under the Construct menu. o Click in a blank space to deselect everything. Select the right endpoint (point B) and the original segment (segment AB). Select Perpendicular Line under the Construct menu. o Using the selection tool click on the precise intersection of the two perpendicular lines to create an intersections point. o Click in a blank space to deselect everything. Highlight the new perpendicular lines. Select Hide Perpendicular Lines under the Display menu. o Using the segment tool, create a segment between Point D and the new intersection point (point E). Also, create a segment between Point B and the new intersection point (point E). This rectangle should have a length to height ratio of 4:3. o Using the segment tool, create a diagonal segment from Point A to Point E. o While the segment is still highlighted, select Length under the Measure menu. o Highlight the vertices of the rectangle in consecutive order (A, B, E, and D). Select Quadrilateral Interior from the Construct menu. o While the quadrilateral interior is still highlighted, select Area under the Measure menu. M. Winking p.29

o Highlight the two measures and select Tabulate under the Graph Menu. This should create an interactive table. o First, alter the rectangle that represents the TV screen by dragging point A or B until it has a diagonal that measures 7cm. Once you have a diagonal that is as close to 7 cm as you can get the measure to be, double click on the table. o What is the area of such a TV screen? o Drag point B again until the area of the TV screen doubles. Again, double click on the interactive table. When the area of the TV screen double did the diagonal measure also double? o If the diagonal did not double, drag point B until the diagonal measures 14 cm which would be double the length of the original TV screen. Again, double click on the interactive table. When the diagonal doubled, what happened to the area? (Did it just double also?) o What happens when you make the diagonal ½ of the original length? What kind of conclusions can you make? o Try highlighting just the table and select Plot Table Data under the Graph Menu. o EXTENSIONS: Try this same lab for a widescreen TV. Wide screen TV s have a ratio of 16:9 for their width to length. What happens with similar solid s and their volumes, compared to surface area, or length of sides? M. Winking p.30

(Picture Ratio s) Ratio s have been used for thousands of years to measure. Thales of Miletus is credit for many of these first practical uses of proportions. The most famous is when he was asked to determine the height of the great pyramids. In this example he used the shadow cast by an object compared to the objects height. 10 803 6 x There are many other similar uses of proportions and ratios that can be demonstrated in Geometer s Sketchpad. Have students set up a mirror on the floor and locate the top of a tall object in the mirror as shown in the diagram below. Reflections create an angle of incidence and angle of reflection which are congruent. The base of the flagpole and the ground make a right angle as well as the girl and the ground make a right angle. So, there are two congruent interior angles between the two triangles and must therefore be similar triangles which suggest corresponding sides are proportional. You can use a digital camera to take a picture of a similar situation and paste the picture into Sketchpad to investigate the similar proportions. Sometimes perspectives of pictures can be misleading. To minimize the effect of perspectives, try to take the picture on the perpendicular bisector of the line between the person looking into the mirror and the flagpole. 62 in. 21 in. 72 in. mirror o Find an object, set up the mirror, the person, and take a picture as recommended above. Measure the person s distance to the mirror, the distance form the mirror to the flagpole, and the height of the person to their eyes. Finally, open the picture on the computer. Then, copy and paste the picture into Sketchpad. o Right click on the picture and select Properties o Make sure to uncheck the radio button next to Arrow Selectable and press OK. o Now the geometric lines to create the triangles can be superimposed over the actual picture. M. Winking p.31

o In this example the angles of the two triangles with a vertex at the mirror were measured to verify they were congruent. Then, using the labeling tool a little text box was created and measurements added. o Using a simple ratio would enable us to find a reasonable height for the sign. 1.2 1.5 7.3 x 1.2 1.5 7.3 x 10.95 1.2x 10.95 1.2x 1.2 1.2 x 9. 125m o We can also estimate the height of the of the sign using the ratios in the picture. Highlight the segment representing the person and the sign. Select Length under the Measure menu. This picture shows that 1.48 cm 1.5 m. This suggests the following ratio could be setup. 1.48 9.74 1.5 x 1.48 9.74 1.5 x 14.61 1.48x 14.61 1.48x 1.48 1.48 x 9. 87m The answer is still reasonably close to the first answer. M. Winking p.32