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Here is some recommended reading before getting started.
Try your knowledge on these topics.
Select all the statements that are true.
Without using the Pythagorean Theorem nor the Distance Formula, can it be determined whether the quadrilateral shown below is a rectangle?
In mathematics, there are usually several ways of proving or disproving a statement. The following example will explore two ways of determining whether a point is on a circle.
Ramsha and Dylan are taking Geometry together. They were asked to determine whether the point P(1,3) is on the circle that passes through Q(0,2) and whose center is at the origin. They decided to do it using different methods.The standard equation of the circle with center at (h,k) and radius r is (x−h)2+(y−k)2=r2. The distance d between two points with coordinates (x1,y1) and (x2,y2) is d=(x2−x1)2+(y2−y1)2.
Ramsha has decided to write the standard equation of the circle, and then check whether the coordinates of P satisfy this equation. Conversely, Dylan has decided to use the Distance Formula. These two options will be explored one at a time.
As already said, Ramsha has decided first to write the standard equation of the circle and use it to see whether the point is on that circle. To do so, she will follow two steps.
These steps will be done one at a time.
As previously stated, Dylan has decided to use the Distance Formula. To do so, he will follow two steps.
These steps will be done one at a time.
Real life problems can also be solved by setting a coordinate plane and using geometric properties.
A small country has three main roads connecting the beach, the mountains, and a national park. To avoid traffic jams and car accidents during summer holidays, the government of the country is planning to build a fourth road to connect the midpoints of Road 1 and Road 2.
After placing the diagram on a coordinate plane, use the Midpoint Formula to find the coordinates of the midpoints of Road 1 and Road 2.
Midpoint Formula: (2x1+x2,2y1+y2) | |||
---|---|---|---|
Road | Points | Substitute | Simplify |
Road 1 | B(4,0) and M(-4,4) | M1(24+(-4),20+4) | M1(0,2) |
Road 2 | M(-4,4) and P(-2,-2) | M2(2-4+(-2),24+(-2)) | M2(-3,1) |
Distance Formula: d=(x2−x1)2+(y2−y1)2 | |||
---|---|---|---|
Points | Substitute | Simplify | |
M1(0,2) and M2(-3,1) | dM1M2=(-3−0)2+(1−2)2 | dM1M2=10 | |
B(4,0) and P(-2,-2) | dBP=(-2−4)2+(-2−0)2 | dBP=210 |
Slope Formula: m=x2−x1y2−y1 | |||
---|---|---|---|
Points | Substitute | Simplify | |
M1(0,2) and M2(-3,1) | mM1M2=-3−01−2 | mM1M2=31 | |
B(4,0) and P(-2,-2) | mBP=-2−4-2−0 | mBP=31 |
The result obtained in the previous example can be generalized to any triangle.
DE∥BC and DE=21BC
This theorem can be proven by placing the triangle on a coordinate plane. For simplicity, vertex B will be placed at the origin and vertex C on the x-axis.
Since B lies on the origin, its coordinates are (0,0). Point C is on the x-axis, meaning its y-coordinate is 0. The remaining coordinates are unknown and can be named a, b, and c.To prove this theorem, it must be proven that DE is parallel to BC and that DE is half of BC.
If the slopes of these two segments are equal, then they are parallel. The y-coordinate of both B(0,0) and C(a,0) is 0. Therefore, BC is a horizontal segment. Next, the coordinates of D and E will be found using the Midpoint Formula.
M(2x1+x2,2y1+y2) | |||
---|---|---|---|
Segment | Endpoints | Substitute | Simplify |
BA | B(0,0) and A(b,c) | D(20+b,20+c) | D(2b,2c) |
CA | C(a,0) and A(b,c) | E(2a+b,20+c) | E(2a+b,2c) |
Since both BC and DE are horizontal, their lengths are given by the difference of the x-coordinates of their endpoints.
Segment | Endpoints | Length | Simplify |
---|---|---|---|
BC | B(0,0) and C(a,0) | BC=a−0 | BC=a |
DE | D(2b,2c) and E(2a+b,2c) | DE=2a+b−2b | DE=21a |
Not only can coordinates be used to prove properties of triangles, but also they can be used to prove properties of quadrilaterals.
Davontay lives in a neighborhood in Boston where the stadium, the amusement park, the airport, and the train station are the vertices of a quadrilateral that seems to be a parallelogram.See solution.
Place the diagram on a coordinate plane.