Question
Eden is cutting two triangular tiles for her bathroom. She needs the tiles to be congruent but is not sure she is cutting them that way. Eden has ensured that one side of both tiles is congruent. Which pair of sides would Eden need to compare in order to make sure the triangles are congruent by HL?
triangle ABC with coordinates A at 1 comma 4, B at 2 comma 4, C at 1 comma 1, and a right angle symbol at A and length of BC of 3 and 2 tenths units, triangle EFD with coordinates E at 2 comma negative 2, F at one comma negative 2, D at 2 comma 1 and a right angle symbol at E with length of FD of 3 and 2 tenths units
segment AC and segment FD
segment AB and segment EF
segment BC and segment EF
segment BC and segment FD
Answers
GPT-4o mini
To determine whether the triangles ABC and EFD are congruent by the Hypotenuse-Leg (HL) theorem, we need to ensure that we have one pair of congruent sides (the hypotenuses) and one pair of congruent legs.
From the information given:
- Triangle ABC has a right angle at A, and one side (BC) measures 3.2 units.
- Triangle EFD has a right angle at E, and one side (FD) also measures 3.2 units.
Since the hypotenuse (BC) and the leg (FD) of both triangles are congruent, we can use the HL theorem.
Now, we need to identify the congruent leg. Given that we already have the hypotenuse as BC and FD, we need to compare another leg from each triangle.
The pairs of sides available for comparison are:
- Segment AC (which is a leg of triangle ABC)
- Segment AB (which is a leg of triangle ABC)
- Segment EF (which is a leg of triangle EFD)
- Segment FD (which we've already identified as a hypotenuse)
To apply the HL theorem correctly, we need to identify the lengths of the legs:
1. Calculate the lengths:
- For triangle ABC:
- AC = \( \sqrt{(1 - 1)^2 + (1 - 4)^2} = \sqrt{0 + 9} = 3 \) units
- AB = \( \sqrt{(2 - 1)^2 + (4 - 4)^2} = \sqrt{1 + 0} = 1 \) unit
- For triangle EFD:
- EF = \( \sqrt{(1 - 2)^2 + (-2 + 2)^2} = \sqrt{1 + 0} = 1 \) unit
- ED = \( \sqrt{(2 - 2)^2 + (1 + 2)^2} = \sqrt{0 + 9} = 3 \) units
2. Identify congruent legs:
- Segment AB (1 unit) and segment EF (1 unit) are congruent.
Thus, the correct pairs that Eden needs to compare are:
- **Segment AB and segment EF**
This comparison will confirm that the triangles are congruent by HL, as both triangles have two congruent sides: one hypotenuse (BC and FD) and one leg (AB and EF).
From the information given:
- Triangle ABC has a right angle at A, and one side (BC) measures 3.2 units.
- Triangle EFD has a right angle at E, and one side (FD) also measures 3.2 units.
Since the hypotenuse (BC) and the leg (FD) of both triangles are congruent, we can use the HL theorem.
Now, we need to identify the congruent leg. Given that we already have the hypotenuse as BC and FD, we need to compare another leg from each triangle.
The pairs of sides available for comparison are:
- Segment AC (which is a leg of triangle ABC)
- Segment AB (which is a leg of triangle ABC)
- Segment EF (which is a leg of triangle EFD)
- Segment FD (which we've already identified as a hypotenuse)
To apply the HL theorem correctly, we need to identify the lengths of the legs:
1. Calculate the lengths:
- For triangle ABC:
- AC = \( \sqrt{(1 - 1)^2 + (1 - 4)^2} = \sqrt{0 + 9} = 3 \) units
- AB = \( \sqrt{(2 - 1)^2 + (4 - 4)^2} = \sqrt{1 + 0} = 1 \) unit
- For triangle EFD:
- EF = \( \sqrt{(1 - 2)^2 + (-2 + 2)^2} = \sqrt{1 + 0} = 1 \) unit
- ED = \( \sqrt{(2 - 2)^2 + (1 + 2)^2} = \sqrt{0 + 9} = 3 \) units
2. Identify congruent legs:
- Segment AB (1 unit) and segment EF (1 unit) are congruent.
Thus, the correct pairs that Eden needs to compare are:
- **Segment AB and segment EF**
This comparison will confirm that the triangles are congruent by HL, as both triangles have two congruent sides: one hypotenuse (BC and FD) and one leg (AB and EF).