If the equation y=mx+b is used to model a quantity y as a function of the quantity x, why is b considered to be the starting val
1 answer:
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The midpoint theorem states that the line joining the mid points of two sides of a triangle is parallel to the third and facing side and equal to half of the length of the third side
Based on the midpoint theorem, the area of triangle ΔLPQ is 63 square units
The reason the value of the area of triangle ΔLPQ as given above is correct is as follows:
The given parameters;
The midpoint of = M; The midpoint of
= N; The midpoint of
= O
The midpoint of = P; The midpoint of
= Q; The midpoint of
= R
The area of triangle ΔPQR = 21
The required parameter:
Calculate the area of triangle ΔLPQ
Method:
The definition of midpoint, area ratio, and area of a triangle formula can be used to find the area of triangle ΔLPQ
Solution:
According to the midpoint theorem, we have;
= (1/2) ×
= (1/2) ×
= (1/2) ×
Given that is parallel to
, and
is parallel to, we have;
∠MON = ∠PRQ
Similarly, we have, ∠MNO = ∠PQR
Therefore, ΔPQR is similar to triangle ΔMON, which is also similar to ΔJKS
The area of triangle ΔPQR = 21, by area ratio = (Side ratio)², we have;
The sides of ΔMON = 2 × The side length of ΔPQR
The area of triangle ΔMON = 2² × The area of ΔPQR
∴ The area of triangle ΔMON = 4 × 21
Similarly the area of ΔJKS = 4 × 4 × 21
PQ = JK/4
The area of LPQ = (1/2) × PQ × h
h = (3/4×JL) × sin(x°)
∴ The area of LPQ = (1/2)×JK/4×(3/4×JL) × sin(x°)
However; (1/2)×JK×JL× sin(x°) = Area of ΔJKS = 4 × 4 × 21
Therefore;
The area of ΔLPQ = (Area of ΔJKS)/4×(3/4) = (4 × 4 × 21)/4×(3/4) = 63
The area of triangle ΔLPQ = 63 square units
Learn more about the midpoint theorem here:
Answer:
C. True; by the Invertible Matrix Theorem if the equation Ax=0 has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the n x n identity matrix.
Step-by-step explanation:
The Invertible matrix Theorem is a Theorem which gives a list of equivalent conditions for an n X n matrix to have an inverse. For the sake of this question, we would look at only the conditions needed to answer the question.
- There is an n×n matrix C such that CA=
.
- There is an n×n matrix D such that AD=
- The equation Ax=0 has only the trivial solution x=0.
- A is row-equivalent to the n×n identity matrix
- For each column vector b in
, the equation Ax=b has a unique solution.
- The columns of A span
Therefore the statement:
If there is an n X n matrix D such that AD=I, then there is also an n X n matrix C such that CA = I is true by the conditions for invertibility of matrix:
- The equation Ax=0 has only the trivial solution x=0.
- A is row-equivalent to the n×n identity matrix
The correct option is C.