Note: For questions 13–21, remember to show all of the steps that you use to solve the problem. You can use the comments field to explain our work. Your teacher will review each step of your responses to ensure your receive proper credit for your answers.
Note: Your teacher will grade your response to this question to ensure you receive proper credit for your answer.
Complete the proof.
Given: AB · BE = CB · BD
Prove: ΔABC ~ ΔDBE
To prove that ΔABC ~ ΔDBE, we need to show that all corresponding angles are congruent and all corresponding sides are proportional.
We know that AB · BE = CB · BD, so we can set up a proportion:
AB/CB = BD/BE
This tells us that the ratio of the corresponding sides AB and CB is equal to the ratio of the corresponding sides BD and BE.
Now, let's look at the corresponding angles:
Angle A is congruent to angle D (vertical angles)
Angle B is congruent to angle E (vertical angles)
Angle C is congruent to angle BDC (alternate interior angles)
Therefore, all corresponding angles are congruent and all corresponding sides are proportional. This proves that ΔABC ~ ΔDBE.
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Note: Your teacher will grade your response to ensure you receive proper credit for your answer. The box-and-whisker plot below shows the numbers of text messages received in one day by students in the seventh and eighth grades at Lincoln Middle School. Two box and whisker plots are shown above a number line. The number line is scaled from 2 to 38 by twos. The first plot above the number line is labeled seventh grade. On the seventh grade plot: The left point is at 6. The left side of the box is at 14. The vertical segment inside the box is at 18 . The right side of the box is at 22. The right point of the plot is at 26. The eighth grade plot is above the seventh grade plot. On the eighth grade plot: The left point is at 22. The left side of the box is at 26. the vertical segment inside the box is at 30. The right side of the box is at 34. The right point of the plot is at 38. Where do the two sets of data overlap? Find the IQR of each set. Find the difference between the medians of the sets. What number multiplied by the IQR equals the difference between the medians of the two sets?
The two sets of data overlap between 22 and 26.
To find the IQR of the seventh grade set:
IQR = Q3 - Q1 = 22 - 14 = 8
To find the IQR of the eighth grade set:
IQR = Q3 - Q1 = 34 - 26 = 8
The difference between the medians of the two sets is:
Median of seventh grade set = (18 + 22)/2 = 20
Median of eighth grade set = (30 + 34)/2 = 32
Difference in medians = 32 - 20 = 12
Let x be the number multiplied by the IQR to get the difference in medians:
x(IQR) = 12
x = 12/8 = 1.5
Therefore, the number multiplied by the IQR to get the difference between the medians of the two sets is 1.5.
Note: Your teacher will grade your response to ensure you receive proper credit for your answer. The box-and-whisker plot below shows the numbers of text messages received in one day by students in the seventh and eighth grades at Lincoln Middle School. Two box and whisker plots are shown above a number line. The number line is scaled from 2 to 38 by twos. The first plot above the number line is labeled seventh grade. On the seventh grade plot: The left point is at 6. The left side of the box is at 14. The vertical segment inside the box is at 18 . The right side of the box is at 22. The right point of the plot is at 26. The eighth grade plot is above the seventh grade plot. On the eighth grade plot: The left point is at 22. The left side of the box is at 26. the vertical segment inside the box is at 30. The right side of the box is at 34. The right point of the plot is at 38. Where do the two sets of data overlap? Find the IQR of each set. Find the difference between the medians of the sets. What number multiplied by the IQR equals the difference between the medians of the two sets?
The two sets of data overlap between 22 and 26.
To find the IQR of the seventh grade set:
IQR = Q3 - Q1 = 22 - 14 = 8
To find the IQR of the eighth grade set:
IQR = Q3 - Q1 = 34 - 26 = 8
The difference between the medians of the two sets is:
Median of seventh grade set = (18 + 22)/2 = 20
Median of eighth grade set = (30 + 34)/2 = 32
Difference in medians = 32 - 20 = 12
Let x be the number multiplied by the IQR to get the difference in medians:
x(IQR) = 12
x = 12/8 = 1.5
Therefore, the number multiplied by the IQR to get the difference between the medians of the two sets is 1.5.
A class of 24 students was asked to identify a team sport in which they have participated. Their responses are shown in the line plot.
A line plot is shown for 5 sports.
The first sport is football. Above football there are 4 x marks.
The second sport is soccer. Above soccer there are 8 x marks.
The third sport is baseball. Above baseball there are 7 x marks.
The fourth sport is basketball. Above basketball there are 5 x marks.
The fifth sport is Track and Field. Above Track and Field there are 4 x marks.
How many total responses are recorded in the line plot for the five sports?
What is the total number of students polled?
Give a possible explanation as to why the answers to parts (a) and (b) are different.
(a) To find the total number of responses recorded in the line plot, we need to add up the number of x marks for each sport:
Number of responses for football = 4
Number of responses for soccer = 8
Number of responses for baseball = 7
Number of responses for basketball = 5
Number of responses for Track and Field = 4
Total number of responses = 4 + 8 + 7 + 5 + 4 = 28
(b) The total number of students polled is 24, since that is the number of students in the class who were asked to identify a team sport in which they have participated.
(c) The total number of responses (28) is greater than the total number of students polled (24) because some students may have participated in more than one team sport, and therefore would be counted multiple times in the line plot. For example, a student who played both soccer and basketball would be counted twice in the line plot, once above the soccer category and once above the basketball category.
To prove that the triangles ΔABC and ΔDBE are similar, we need to show that their corresponding angles are congruent and their corresponding sides are in proportion.
Given that AB · BE = CB · BD, we can utilize the proportionality of sides and angle-side-angle similarity criterion to prove the similarity of the triangles.
Proof:
Step 1: Draw a diagram representing the given triangles ΔABC and ΔDBE.
Step 2: Since AB · BE = CB · BD, we can write the proportionality relationship as follows: AB/BD = CB/BE.
Step 3: Consider angle A in ΔABC and angle D in ΔDBE. These angles are corresponding angles, as they are opposite sides of the parallelogram ABCD. Therefore, angle A ≅ angle D.
Step 4: Using the proportionality relationship AB/BD = CB/BE, we can rewrite it as AB/BC = BD/BE.
Step 5: Here, we have angle A ≅ angle D and AB/BC = BD/BE. By the angle-side-angle similarity criterion, we can conclude that ΔABC ~ ΔDBE.
Thus, we have proven that the triangles ΔABC and ΔDBE are similar.