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Here are a few recommended readings before getting started.
Try your knowledge on these topics.
The figure below looks like a parallelogram.
Without using measuring tools such as a ruler or a protractor, how can it be proven that the quadrilateral above is actually a parallelogram?In certain situations, it is required to find the length of a line segment or a side of a polygon. To find those lengths, it is recommended to calculate the distance between the endpoints of the segment, or between two vertices of a polygon. If those points are plotted on a coordinate plane, the Distance Formula can be used.
Given two points A(x1,y1) and B(x2,y2) on a coordinate plane, their distance d is given by the following formula.
Start by plotting A(x1,y1) and B(x2,y2) on the coordinate plane. Both points can be arbitrarily plotted in Quadrant I for simplicity. Note that the position of the points in the plane does not affect the proof. Assume that x2 is greater than x1 and that y2 is greater than y1.
Next, draw a right triangle. The hypotenuse of this triangle will be the segment that connects points A and B.a=x2−x1, b=y2−y1
Use the Distance Formula to calculate the distance between the points plotted in the coordinate plane. If needed, round the answer to two decimal places.
Ramsha and Davontay are learning to make picture frames so they want to determine whether the quadrilateral ABCD shown below is a rectangle.
Ramsha says that ABCD is a rectangle. Davontay says it is not. Use the Distance Formula to decide who is correct.Other than just using the Distance Formula, try the Converse of the Pythagorean Theorem to determine whether the angles of the quadrilateral are right angles.
d=(x2−x1)2+(y2−y1)2 | |||
---|---|---|---|
Segment | Points | Substitute | Simplify |
AC | A(-3,0) & C(4,-1) | AC= (4−(-3))2+(-1−0)2 | AC= 52 |
AB | A(-3,0) & B(-2,2) | AB= (-2−(-3))2+(2−0)2 | AB= 5 |
BC | B(-2,2) & C(4,-1) | BC= (4−(-2))2+(-1−2)2 | BC= 35 |
By following the same procedure, it can be shown that ∠A, ∠C, and ∠D are also right angles.
Since each angle of ABCD is a right angle, the quadrilateral is a rectangle. Therefore, Ramsha is correct. She's on her way toward making some nice frames.
Start by finding the radius of the circle.
Sometimes the length of a segment is not what is needed. Instead, the coordinates of the point that lies exactly in the middle of a segment are needed.
If the points are plotted on a coordinate plane, the Midpoint Formula can be used to find the coordinates of the midpoint.
The midpoint M between two points A(x1,y1) and B(x2,y2) on a coordinate plane can be determined by the following formula.
M(2x1+x2,2y1+y2)
The formula above is called the Midpoint Formula.
LHS+xm=RHS+xm
LHS+x1=RHS+x1
Commutative Property of Addition
LHS/2=RHS/2
M(2x1+x