Answer:
Explanation:
Given that AABC~AADE, we know that the corresponding angles in the two triangles are equal, and the corresponding sides are proportional.
a. To find the scale factor from AABC to AADE, we can compare the lengths of corresponding sides. Specifically, we can compare the length of AB to the length of AD, since these sides correspond to each other in the two triangles.
We have:
AB/AD = BC/DE (since the triangles are similar)
Substituting the given values, we get:
10/x = 21/12
x = 12(10)/21
x = 40/7
Therefore, the scale factor from AABC to AADE is 40/7.
b. To find the value of x, we used the fact that AB/AD = BC/DE. Substituting the given values, we obtained x = 40/7.
c. To find m/ABC, we can use the fact that corresponding angles in similar triangles are equal. We know that AABC~AADE, so we have:
m/ABC = m/AAE
We are given that m/AAE = 32°, so m/ABC = 32°.
d. To find the perimeter of AADE, we can use the scale factor we found in part (a) to find the lengths of the corresponding sides in AADE. Specifically, we have:
AD = AB * (40/7)
AD = 10 * (40/7)
AD = 400/7
DE = BC * (40/7)
DE = 21 * (40/7)
DE = 840/7
AE = AB + AD = 10 + 400/7
AE = 107.14...
DE + EA + AD = 840/7 + 107.14... + 400/7 = 1200/7
Therefore, the perimeter of AADE is approximately 171.43 units.
e. To find the area of AADE, we can use the fact that the area of similar triangles is proportional to the square of the scale factor. Specifically, we have:
Area of AABC / Area of AADE = (AB/AD)^2
Substituting the given values and the value of x we found earlier, we get:
Area of AABC / Area of AADE = (10/AD)^2
71.75 / Area of AADE = (10/(40/7))^2
Area of AADE = 71.75 * (40/7)^2 / 100
Area of AADE = 122.29 square units (approx)
Therefore, the area of AADE is approximately 122.29 square units.
f. To determine whether BC || DE, we need to look at the corresponding angles in the two triangles. We know that AABC~AADE, so the corresponding angles are equal.
In particular, we know that m/ABC = m/AAE and m/ABD = m/ADE. We found that m/ABC = 32° in part (c), and we can see from the diagram that m/ABD = 180° - m/ABC - m/BAC = 180° - 32° - 40° = 108°.
Therefore, we have:
m/ADE = m/ABD = 108°
We can also see from the diagram that m/AED = m/AEB + m/DEC = 40° + 32° = 72°.
Therefore, we have:
m/AED + m/ADE = 72° + 108° = 180°