<u>Correct </u><u>Inputs </u><u>:-</u>
In ΔABC right angled at A, D and E are points on BC, C such that BD = CD and AD ⊥ BC

Let us know about definition of altitude first. The altitude of a triangle is the perpendicular line segment drawn from the vertex to the opposite side of the triangle.
Median is the line segment from a vertex to the midpoint of the opposite side.
<u>Let us Check all options one by one </u>
- CD is line segment which starts from vertex C but don't falls on opposite side AB thus it is not an altitude.❌
- BA is line segment which starts from vertex B and falls perpendicularly on opposite sides AC and is thus an altitude.✔️
- AD is line segment which starts from vertex A and falls perpendicularly on opposite side BC and is thus an altitude.✔️
- AE is a line segment which starts from vertex A but doesn't falls perpendicularly on opposite side BC and is thus not an altitude.❌
- AD falls on BC with D as mid point because BD = CD and is thus a median. ✔️
Answer:
Part c: Contained within the explanation
Part b: gcd(1200,560)=80
Part a: q=-6 r=1
Step-by-step explanation:
I will start with c and work my way up:
Part c:
Proof:
We want to shoe that bL=a+c for some integer L given:
bM=a for some integer M and bK=c for some integer K.
If a=bM and c=bK,
then a+c=bM+bK.
a+c=bM+bK
a+c=b(M+K) by factoring using distributive property
Now we have what we wanted to prove since integers are closed under addition. M+K is an integer since M and K are integers.
So L=M+K in bL=a+c.
We have shown b|(a+c) given b|a and b|c.
//
Part b:
We are going to use Euclidean's Algorithm.
Start with bigger number and see how much smaller number goes into it:
1200=2(560)+80
560=80(7)
This implies the remainder before the remainder is 0 is the greatest common factor of 1200 and 560. So the greatest common factor of 1200 and 560 is 80.
Part a:
Find q and r such that:
-65=q(11)+r
We want to find q and r such that they satisfy the division algorithm.
r is suppose to be a positive integer less than 11.
So q=-6 gives:
-65=(-6)(11)+r
-65=-66+r
So r=1 since r=-65+66.
So q=-6 while r=1.
Answer to the miscellaneous equation, is x=0
Miscellaneous equation are the equations which are not polynomial.
The question can be solved by:
Factorising:Splitting the terms to find the required solution
Completing the squares, etc
2ˣ -3ˣ=√(6ˣ-9ˣ)
-> (2ˣ -3ˣ)²=(3ˣ.2ˣ - 3²ˣ)
Squaring both sides,
-> 2²ˣ+3²ˣ- 2.3ˣ.2ˣ=(3ˣ.2ˣ - 3²ˣ)
-> 2²ˣ+2.3²ˣ-2.3ˣ.2ˣ-3ˣ.2ˣ=0
Factorising the terms,
-> 2ˣ(2ˣ-3ˣ) -2.3ˣ(2ˣ-3ˣ)=0
-> (2ˣ-3ˣ)(2ˣ-2.3ˣ)=0
Equating the braces to zero,
x=0 is the only solution.
Therefore, x =0 is the only solution
Learn more about miscellaneous equation: brainly.com/question/1214333
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