Question

DE is a perpendicular bisector of FG, and they intersect at point H. which of the following statements must be true? select the two correct answers.
a. DH = EH
b. FH = GH
c. FG = DE
d. DHF = DHG
e. FDH = DGH

Answer 1

Given that DE is the perpendicular bisector of FG and they intersect at point H, we can analyze the statements:

1. **Statement a: DH = EH**
This statement is true. Since DE is the perpendicular bisector, it divides FG into two equal segments at H. Therefore, DH is equal to EH.

2. **Statement b: FH = GH**
This statement is also true. By definition of a bisector, FH and GH are equal because H is the midpoint of FG.

3. **Statement c: FG = DE**
This statement is not necessarily true. The lengths of FG and DE can differ; the perpendicular bisector does not imply that those segments are equal in length.

4. **Statement d: DHF = DHG**
This statement is not true. The angles DHF and DHG may be equal because DE is perpendicular to FG; however, we are not specifically provided with a relationship that confirms their equality as angles.

5. **Statement e: FDH = DGH**
This statement is not necessarily true. The angles FDH and DGH are not guaranteed to be equal without further information about the configuration.

Therefore, the two correct statements that must be true are:
1. **a. DH = EH**
2. **b. FH = GH**