Mid – Module Assessment Review

Example 1 Given center 𝑂 and quadrilateral 𝐴𝐵𝐶𝐷, using a compass and ruler, dilate the figure from center 𝑂 by a scale factor of 𝑟=2. Label the dilated quadrilateral 𝐴′𝐵′𝐶′𝐷′.

Example 1 Given center 𝑂 and quadrilateral 𝐴𝐵𝐶𝐷, using a compass and ruler, dilate the figure from center 𝑂 by a scale factor of 𝑟=2. Label the dilated quadrilateral 𝐴′𝐵′𝐶′𝐷′. Step 1: Use a straight edge to draw lines from point O through each of the points A, B, C, D. Step 2: Measure the distance from point O to point A using your compass. Step 3: Copy that distance on the line you drew from point O to through point A by moving the round center of the compass up to point A and marking the distance on the line. Label that point A’. Step 4: Repeat step 3 for points B, C, and D. Step 5: Connect the dots in the same order as the original figure.

Example 1 a. Your image should look something like this. Note: To draw a second image that has a scale factor of 𝑟= 1 2 , Use a ruler and measure each distance from O to each point A, B, C, D. Then cut that distance in half and mark the point on the lines that you’ve drawn. The new image should be between point O and the original figure and will be half the size.

Example 2 Let 𝐷 be the dilation with center 𝑂 and scale factor 𝑟>0 so that 𝐷𝑖𝑙𝑎𝑡𝑖𝑜𝑛 𝑃 =𝑃′ and 𝐷𝑖𝑙𝑎𝑡𝑖𝑜𝑛 𝑄 = 𝑄 ′ . a. Use lengths 𝑂𝑄 =15 units and 𝑂 𝑄 ′ =25 units to determine the scale factor 𝑟 of dilation 𝐷. Describe how to determine the coordinates of 𝑃′ using the coordinates of 𝑃. Using the definition of a dilation, 𝑂𝑄′ =𝑟 𝑂𝑄 , we have that 25=𝑟∙15. Solving for r we get 𝑟= 25 15 = 5 3 . To find the coordinates of point P’ we simply multiply the coordinates of P by our scale factor 5 3 . Since 𝑃= −4, −3 , 𝑃 ′ = −4× 5 3 , −3× 5 3 = − 20 3 , −5 b. If 𝑂𝑄 =15 units, 𝑂 𝑄 ′ =25 units, and 𝑃 ′ 𝑄 ′ =12.5 units, determine the length of 𝑃𝑄 . Round your answer to the tenths place, if necessary. Since we know the definition of dilation is 𝑃′𝑄′ =𝑟 𝑃𝑄 and we know 𝑟= 5 3 , we can substitute the values 𝑃 ′ 𝑄′ =12.5 and 𝑟= 5 3 into the equation and solve for 𝑃𝑄 . So we get 12.5= 5 3 ∙ 𝑃𝑄 , multiplying both sides by 3 5 gives us 𝑃𝑄 =7.5.

Example 3 Is there a dilation from center O that would map ∆𝐴𝐵𝐶 onto ∆ 𝐴 ′ 𝐵 ′ 𝐶′? If yes describe the dilation in terms of coordinates of corresponding points. The distance from the x – axis to point C is 12 units. The distance from the x – axis to point C’ is 4 units. That is a ratio of 4 12 = 1 3 . Similarly, the distance from the x – axis to point B is 3 units and the distance from the x – axis to point B’ is 1 unit. That is also a ratio of 1 3 . Therefore there is a dilation of ∆𝐴𝐵𝐶 by a scale factor of 𝑟= 1 3 that would map ∆𝐴𝐵𝐶 onto ∆ 𝐴 ′ 𝐵 ′ 𝐶′. Additionally: 𝐴 ′ = 1 3 ×3, 1 3 ×2 = 1, 2 3 𝐵 ′ = 1 3 ×12, 1 3 ×3 = 4, 1 𝐶 ′ = 1 3 ×9, 1 3 ×12 = 3, 4

Example 4 Triangle 𝐴𝐵𝐶 is located at points 𝐴= −2, 3 , 𝐵= 2, 2 , and 𝐶= 2, 4 and has been dilated from the origin by a scale factor of 3. Draw and label the vertices of triangle 𝐴𝐵𝐶. Determine the coordinates of the dilated triangle 𝐴 ′ 𝐵 ′ 𝐶 ′ , and draw and label it on the coordinate plane. The coordinates of triangle 𝐴 ′ 𝐵 ′ 𝐶′ are: 𝐴 ′ = 3× −2 , 3×3 = −6, 9 𝐵 ′ = 3×2, 3×2 = 6, 6 𝐶 ′ = 3×2, 3×4 = 6, 12

Example 4 continued…