CD is a perpendicular bisector of AB, and they intersect at point M.

Question

Which of the following statements must be true? Choose two
angle DMA= angle DMB
segment CM= segment DM
segment AD= segment CD
Angle BDM= Angle CMA
segment AM= segment BM

Answer 1

Given that CD is a perpendicular bisector of AB that intersects at point M, we can analyze the properties of the triangle formed and the relationships between the segments and angles involved.

1. Angle DMA = Angle DMB: This is true because CD is the perpendicular bisector of AB; thus, angles DMA and DMB are formed on opposite sides of the line CD and are equal.

2. Segment CM = Segment DM: This statement does not have to be true since we do not have information that relates segments CM and DM being equal based on the information given.

3. Segment AD = Segment CD: This statement is not necessarily true as there is no information provided that suggests this relationship.

4. Angle BDM = Angle CMA: This statement is not necessarily true - angles BDM and CMA are not assured to be equal based on the information given.

5. Segment AM = Segment BM: This must be true because M is the midpoint of AB, as CD bisects AB. Therefore, segments AM and BM are equal.

Thus, the two statements that must be true are:

1. Angle DMA = Angle DMB
2. Segment AM = Segment BM

Answer 2

The figure shown is a rectangle. If BD = x + 15 and AC = 3x + 1, what is the total length of

Answer 3

In a rectangle, the diagonals are equal in length. Thus, if we have \( BD = AC \). Given:

• \( BD = x + 15 \)
• \( AC = 3x + 1 \)

We can set these two expressions equal to each other to solve for \( x \):

\[ x + 15 = 3x + 1 \]

Now, let's solve for \( x \):

1. Subtract \( x \) from both sides:

\[ 15 = 2x + 1 \]

2. Subtract 1 from both sides:

\[ 14 = 2x \]

3. Divide both sides by 2:

\[ x = 7 \]

Now that we have \( x \), we can find the lengths of \( BD \) and \( AC \):

Substituting \( x \) into \( BD \):

\[ BD = 7 + 15 = 22 \]

Substituting \( x \) into \( AC \):

\[ AC = 3(7) + 1 = 21 + 1 = 22 \]

The total length of either diagonal \( BD \) or \( AC \) is therefore \( 22 \).

The total length of the diagonals of the rectangle \( AC \) and \( BD \) would simply be \( 22 + 22 = 44 \) if that's what you're asking for. Otherwise, the length of each diagonal, which is the desired final answer, is \( 22 \).

Thus:

1. Length of each diagonal: \( 22 \)
2. Total length of diagonals: \( 44 \) (if both are added together).