Math  /  Geometry

QuestionWhich descriptions from the list below accurately describe the relationship between ABC\triangle A B C and EDEF\triangle_{E} D E F ? Check all that apply. A. Same area B. Same size C. Congruent D. Congruent after a dilation

Studdy Solution

STEP 1

1. ABC\triangle ABC and DEF\triangle DEF are right triangles.
2. ABC\triangle ABC has sides of lengths 3, 4, and 5, and angles of 37° and 53°.
3. DEF\triangle DEF has sides of lengths 6, 8, and 10, and angles of 37° and 53°.
4. The problem involves checking the properties of area, size, congruence, and dilation congruence.

STEP 2

1. Calculate and compare the areas of ABC\triangle ABC and DEF\triangle DEF.
2. Compare the sizes of ABC\triangle ABC and DEF\triangle DEF.
3. Determine if ABC\triangle ABC and DEF\triangle DEF are congruent.
4. Determine if ABC\triangle ABC and DEF\triangle DEF are congruent after a dilation.

STEP 3

Calculate the area of ABC\triangle ABC using the formula for the area of a right triangle: AreaABC=12×base×height=12×3×4=6 \text{Area}_{ABC} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 4 = 6

STEP 4

Calculate the area of DEF\triangle DEF using the formula for the area of a right triangle: AreaDEF=12×base×height=12×6×8=24 \text{Area}_{DEF} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 8 = 24

STEP 5

Compare the areas of ABC\triangle ABC and DEF\triangle DEF: AreaABC=6andAreaDEF=24 \text{Area}_{ABC} = 6 \quad \text{and} \quad \text{Area}_{DEF} = 24
Since 6246 \neq 24, the areas are not the same.

STEP 6

Compare the sizes of ABC\triangle ABC and DEF\triangle DEF. Since ABC\triangle ABC has sides 3, 4, 5 and DEF\triangle DEF has sides 6, 8, 10, the sizes are different.

STEP 7

Determine if ABC\triangle ABC and DEF\triangle DEF are congruent. Congruent triangles must have the same side lengths and angles. Given ABC\triangle ABC has sides 3, 4, 5 and DEF\triangle DEF has sides 6, 8, 10, they are not congruent.

STEP 8

Determine if ABC\triangle ABC and DEF\triangle DEF are congruent after a dilation. A dilation changes the size but not the shape, meaning the angles remain the same and the side lengths are proportional.
The side lengths of DEF\triangle DEF are exactly twice those of ABC\triangle ABC: 63=84=105=2 \frac{6}{3} = \frac{8}{4} = \frac{10}{5} = 2 Since the side lengths are proportional and the angles are the same (37° and 53°), DEF\triangle DEF is congruent to ABC\triangle ABC after a dilation by a factor of 2.
Solution: - A. Same area: False - B. Same size: False - C. Congruent: False - D. Congruent after a dilation: True

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