Math Problem Statement
Question 2 (Multiple Choice Worth 1 Points) (02.03 MC)
If ΔABC ≅ ΔEDF where the coordinates of A(−1, 1), B(2, 4), and C(3, 1), what is the measure of EF? 3 3.2 4 4.6 Question 3 (Multiple Choice Worth 1 Points) (02.03 MC)
Are triangles ABC and ADC congruent?
triangles ABC and ADC share segment AC, triangle ABC has one dash on side AB and two dashes on side BC, triangle ADC has one dash at side AD and two dashes on side DC Yes, by SSS Yes, by ASA Yes, by SAS Not enough information Question 4 (Multiple Choice Worth 1 Points) (02.03 MC)
What additional information would you need to prove that ΔABC ≅ ΔDEF by ASA?
Triangle ABC and triangle DEF are drawn with angles B and E marked congruent and angles C and F marked congruent. segment AB ≅ segment DE segment BC ≅ segment FE segment AB ≅ segment FE segment BC ≅ segment EF Question 5 (Multiple Choice Worth 1 Points) (02.03 LC)
If ΔFGH ≅ ΔIJK, which segment is congruent to segment FH? segment IK segment JK segment IJ segment FG Question 6 (Multiple Choice Worth 1 Points) (02.03 MC)
If a reflection takes triangle CAT to C′A′T′, what is A′C′?
triangle CAT with vertex A at negative 2 comma 1, vertex T at negative 1 comma 4 and vertex C at 0 comma 0, side AT has a measure of square root of 10 units, side TC has a measure of square root of 17 units, and side AC has a measure of square root of 5 units 3 4 square root of 5 Cannot be determined Question 7 (Multiple Choice Worth 1 Points) (02.03 MC)
Slade draws triangle PQR. He then constructs a perpendicular bisector from vertex P that intersects side QR at point T. What can Slade conclude, based on his drawing? QT = RT TP = RQ PQ = RQ PT = PQ Question 8 (Multiple Choice Worth 1 Points) (02.03 MC)
What coordinate for F would make triangle ABC and triangle DEF congruent?
Triangle ABC is shown. For triangle ABC, A is at 0, 3, B is at negative 1, 2, C is at 2, 3. Two other coordinates are shown. D is at negative 1, 0, E is at 1, negative 2. (−3, −3) (−3, −2) (−2, −3) (−2, −2) Question 9 (Multiple Choice Worth 1 Points) (02.03 LC)
If ΔABC ≅ ΔFDE, which of the following statements is true? ∠A ≅ ∠E ∠B ≅ ∠F ∠A ≅ ∠D ∠B ≅ ∠D Question 10 (Multiple Choice Worth 1 Points) (02.03 LC)
The grid shows Figure Q and its image Figure Q′ after a transformation.
Figure Q is a pentagon drawn on a coordinate grid with vertices in clockwise order at point 2 comma 4, point 3 comma 7, point 7 comma 5, point 5 comma 4, and point 4 comma 2. Figure Q’ is a pentagon drawn with vertices in clockwise order at point negative 2 comma 4, point negative 4 comma 2, point negative 7 comma 3, point negative 5 comma negative 7, and point negative 4 comma 5.
Which of the following transformations could be used to prove that Figure Q and Figure Q′ are congruent? Clockwise rotation of 180° about the origin Counterclockwise rotation of 90° about the origin Clockwise rotation of 90° about the origin Counterclockwise rotation of 270° about the origin
Solution
Let's go through each of the multiple-choice questions one by one.
Question 2:
If ΔABC ≅ ΔEDF, we need to find the measure of segment EF. Since the triangles are congruent, the corresponding sides are equal. We will calculate the length of segment AB, which corresponds to EF.
Using the distance formula to find AB: Thus, the closest answer to this calculation is 4.6.
Answer: 4.6
Question 3:
The question is asking if ΔABC and ΔADC are congruent. Both triangles share segment AC, and based on the markings, both triangles have congruent pairs of sides (AB ≅ AD and BC ≅ DC), and they share side AC.
Since we have two pairs of congruent sides and the included angle, this fits the SAS (Side-Angle-Side) criterion for congruence.
Answer: Yes, by SAS
Question 4:
For proving congruence by ASA (Angle-Side-Angle), you need two angles and the included side to be congruent. The problem states that ∠B ≅ ∠E and ∠C ≅ ∠F. So, to apply the ASA criterion, we need to know that segment BC ≅ EF (the side between the two angles).
Answer: segment BC ≅ segment EF
Question 5:
If ΔFGH ≅ ΔIJK, corresponding sides are congruent. Segment FH corresponds to segment IK in ΔIJK.
Answer: segment IK
Question 6:
A reflection preserves distances between corresponding points in the pre-image and the image. Since the side AC in triangle CAT has a length of √5, the distance A'C' after reflection will also be √5, as reflections preserve lengths.
Answer: √5
Question 7:
Slade constructs a perpendicular bisector from vertex P that intersects side QR at point T. A perpendicular bisector splits a segment into two equal halves, meaning QT = RT.
Answer: QT = RT
Question 8:
To make ΔABC ≅ ΔDEF, corresponding points need to be aligned. By analyzing the given coordinates, we can find that placing F at (−2, −2) will make the triangles congruent.
Answer: (−2, −2)
Question 9:
If ΔABC ≅ ΔFDE, corresponding angles are congruent. From the given choices, ∠A corresponds to ∠D, and ∠B corresponds to ∠F.
Answer: ∠A ≅ ∠D
Question 10:
We are given two figures, Q and Q′, and we need to determine the transformation that maps Q to Q′. By analyzing the points, a clockwise rotation of 180° about the origin will map the vertices correctly between Q and Q′.
Answer: Clockwise rotation of 180° about the origin
If you'd like more details or have questions about any of these answers, feel free to ask!
Here are 5 related questions for further practice:
- How do you apply the distance formula in 3D coordinates?
- What is the difference between the SAS and SSS congruence criteria?
- How do you prove two triangles are congruent using the HL (Hypotenuse-Leg) theorem?
- How do transformations such as reflections affect the orientation of a shape?
- What is the relationship between corresponding angles and sides in congruent polygons?
Tip: When solving geometry problems, always look for congruent angles and sides to identify which congruence postulate applies (SSS, SAS, ASA, etc.).
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Math Problem Analysis
Mathematical Concepts
Congruent Triangles
Coordinate Geometry
Reflection and Transformations
Triangle Theorems
Formulas
Distance formula: d = √((x2 − x1)^2 + (y2 − y1)^2)
Theorems
SAS Congruence Theorem
ASA Congruence Theorem
Reflection Theorem
Suitable Grade Level
Grades 8-10