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!
Draw the angle. Label its vertex O
Place the compass at O and draw a circle that intersects the two rays of the angle. Label the points of intersection A and B. (Extend OA and OB if necessary)
Now, you want angle BOC which shares a side OB with angle AOB, and is congruent to it. It will thus subtend an arc equal to AB. So,
Place the compass at B and draw a circle with radius AB.
It will intersect circle O at A and at another point, labeled C.
angle BOC is adjacent to and congruent with angle AOB.
If you google these exercises, you will find many online diagrams with instructions.
1 To construct an angle that is congruent and adjacent to a given angle, you can follow these steps:
- Start by drawing the given angle using a straightedge.
- Then, place the center of your compass at the vertex of the given angle, and draw an arc that intersects both sides of the angle.
- Without changing the compass opening, place the center of the compass at the point where the angle sides intersect the arc, and draw two more arcs on each side of the angle.
- Finally, use a straightedge to connect the vertex of the given angle with the points where the additional arcs intersect the angle sides. This will construct an angle that is both congruent (same size) and adjacent (shares a common side) to the given angle.
2 When constructing a perpendicular bisector, the compass opening must be greater than half the length of the segment because we need enough room to draw arcs that intersect the segment on both sides. If the compass opening is smaller than half the length of the segment, the arcs would not intersect properly, and the construction would not accurately bisect the segment.
3 When constructing an angle bisector, the arcs must intersect in order to find the exact midpoint of the angle. If the arcs do not intersect, it means that the construction is not precise, and the angle bisector would not be accurate. The intersecting point of the arcs indicates the exact division point of the angle, allowing us to construct the angle bisector correctly.
4 To find the coordinate of the midpoint of a line segment on a number line, you can use the formula:
Midpoint = (Endpoint1 + Endpoint2) / 2
In this case, the endpoints of the line segment are -2 and 12. Plugging them into the formula, we get:
Midpoint = (-2 + 12) / 2 = 10 / 2 = 5
Therefore, the coordinate of the midpoint is 5.
5 To find the coordinates of the other endpoint of a line segment when the midpoint and one endpoint are given, you can use the following formula:
Other Endpoint = (2 * Midpoint) - Endpoint
In this case, the midpoint is E(-1, 0) and one endpoint is C(5, 2). Plugging them into the formula, we get:
Other Endpoint = (2 * (-1), 2 * 0) - (5, 2) = (-2, 0) - (5, 2) = (-2 - 5, 0 - 2) = (-7, -2)
Therefore, the coordinates of the other endpoint are (-7, -2).
6 The distance formula calculates the distance between two points in a coordinate plane. It can be represented as:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
To calculate the distance between points A(1, 1) and B(7, -7), we can plug the coordinates into the distance formula:
Distance = sqrt((7 - 1)^2 + (-7 - 1)^2) = sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10
Therefore, the distance between A(1, 1) and B(7, -7) is 10.