1 Explain how you can use a straightedge and a compass to construct an angle that is both congruent and adjacent to a given angle.
2 When constructing a perpendicular bisector, why must the compass opening be greater than the 1/2 the length of the segment?
3 When constructing an angle bisector, why must the arcs intersect?
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PB is a line segment on a number line. It has endpoints at -2 and 12. What is the coordinate of its midpoint? Please show work!
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5 The midpoint of CD is E(-1,0). One endpoint is C(5,2). What are the coordinates of the other endpoints? Please show work!
6 Explain the distance formula. Then use it to calculate the distance between A(1,1) and B(7,-7). Please show work!
1 To construct an angle that is both congruent and adjacent to a given angle using a straightedge and a compass, follow these steps:
- Place the compass point on the vertex of the given angle and draw an arc that intersects both sides of the angle.
- Without changing the compass opening, place the compass point on one of the points where the arc intersects a side of the given angle and draw another arc inside the angle.
- Use a straightedge to draw a line from the vertex of the given angle to the point where the second arc intersects the angle.
- The constructed angle will be congruent and adjacent to the given angle.
2 When constructing a perpendicular bisector, the compass opening must be greater than half the length of the segment because the midpoint of the segment needs to be determined accurately. By using a compass opening greater than half the length, the arcs drawn on either side of the segment are more likely to intersect, allowing for a more precise construction of the perpendicular bisector.
3 When constructing an angle bisector, the arcs must intersect to ensure that the angle bisector is accurately determined. By drawing arcs from each side of the angle, their intersection will determine the point at which the angle bisector passes through. If the arcs do not intersect, it means that the construction is not accurate, and the angle bisector may not be correct.
4 To find the coordinate of the midpoint of a line segment on a number line, you can use the midpoint formula. The midpoint formula states that the x-coordinate of the midpoint is the average of the x-coordinates of the endpoints, and the same goes for the y-coordinate if applicable.
In this case, PB is a line segment with endpoints at -2 and 12. To find the coordinate of its midpoint, add the x-coordinates and divide by 2:
(-2 + 12) / 2 = 10 / 2 = 5
Therefore, the coordinate of the midpoint is 5.
5 To find the coordinates of the other endpoint of CD when the midpoint is given, you can use the midpoint formula. Following the same logic as explained in question 4, we can find the x-coordinate of the other endpoint:
(x-coordinate of C + x-coordinate of other endpoint) / 2 = x-coordinate of the midpoint
(5 + x-coordinate of other endpoint) / 2 = -1
Solve for x-coordinate of the other endpoint:
5 + x-coordinate of other endpoint = -2
x-coordinate of other endpoint = -2 - 5
x-coordinate of other endpoint = -7
Since the midpoint is given as E(-1,0) and the x-coordinate of the other endpoint is -7, the coordinates of the other endpoint are (-7,0).
6 The distance formula is used to find the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem.
According to the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by the formula:
√((x2 - x1)² + (y2 - y1)²)
To calculate the distance between point A(1,1) and B(7,-7), substitute the values into the formula:
√((7 - 1)² + (-7 - 1)²)
√(6² + (-8)²)
√(36 + 64)
√100
10
Therefore, the distance between A(1,1) and B(7,-7) is 10 units.