Answer:
Step-by-step explanation:
see the attached figure with the letter D to better understand the problem
we know that
The segment side AD is the height of triangle ABC
so
Triangles PBQ and ABD are similar by AA Similarity Theorem
The area of triangle ABC is equal to
we have
substitute
Remember that
If two triangles are similar then the ratio of its corresponding sides is proportional
so
substitute the given values
Answer:
a. {1, 2, 3, 4, 5, 6, 7}
b. {1, 2, 3, 4, 5, 6, 8}
c. {4, 5, 7}
d. {1, 4, 5}
Step-by-step explanation:
The union of two sets is the list of elements in either set. The intersection of two sets is the list of elements in both sets. The complement of a set is the list of elements in the universal set that are not in the set being complemented. The complement of a set can be indicated with an apostrophe: A' is the complement of set A, for example.
<h3>a. AU(BUC)</h3>B∪C = {2, 5, 6, 7} ∪ {3, 4, 6, 7} = {2, 3, 4, 5, 6, 7}
A∪(B∪C) = {1, 4, 5, 7} ∪ {2, 3, 4, 5, 6, 7} = {1, 2, 3, 4, 5, 6, 7}
<h3>b. (AN(BNC))'</h3>B∩C = {2, 5, 6, 7} ∩ {3, 4, 6, 7} = {6, 7}
A ∩ (B∩C) = {1, 4, 5, 7} ∩ {6, 7} = {7}
(A∩(B∩C))' = {7}' = {1, 2, 3, 4, 5, 6, 8}
<h3>c. (ANB)U(ANC)</h3>A∩Β = {1, 4, 5, 7} ∩ {2, 5, 6, 7} = {5, 7}
Α∩C = {1, 4, 5, 7} ∩ {3, 4, 6, 7} = {4, 7}
(A∩B)∪(A∩C) = {5, 7} ∪ {4, 7} = {4, 5, 7}
<h3>d. (ANB')U(ANC')</h3>(A∩B')∪(A∩C') = A∩(B'∪C') = A∩(B∩C)'
(B∩C)' = {6, 7}' = {1, 2, 3, 4, 5, 8}
A∩(B∩C)' = {1, 4, 5, 7} ∩ {1, 2, 3, 4, 5, 8} = {1, 4, 5}
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<em>Additional comment</em>
In part (d) we made use of De Morgan's law for sets:
B'∪C' = (B∩C)'
We also made use of the distributive property for sets:
A∩(B∪C) = (A∩B)∪(A∩C)
