Jason uses the following steps to construct a perpendicular line through a point C on a line segment. Step 1: From point C, draw an arc intersecting the line segment in points A and B. Step 2: Using slightly more compass width, draw two arcs from point A, one above and the other below the line segment. Step 3: Using slightly more compass width, draw two arcs from point B, above and below the line segment. Step 4: Label the point of intersection of the arcs above the line segment as D and below the line as E. Step 5: Using a straightedge, join points D and E. Part A: Which is the first incorrect step? Part B: Using complete sentences, explain your answer for Part A. Part C: Explain why a compass works for the construction done by Jason. (10 points) Quadrilateral ABCD is located at A (-2, 2), B (-2, 4), C (2, 4), and D (2, 2). The quadrilateral is then transformed using the rule (x+2, y-3) to form the image A'B'C'D'. What are the new coordinates of A', B', C', and D'? Describe what characteristics you would find if the corresponding vertices were connected with line segments. (10 points) Triangle ABC is congruent to triangle DEF. In triangle ABC, side AB measures 6, side BC measures 5x-21, and side CA measures 5. In triangle DEF, side DE measures 6, side EF measures x-1, and side FD measures 5. What equation would help you to solve for the side length of BC and EF? Explain your reasoning using complete sentences. (10 points) Write an indirect proof to show that a rectangle has congruent diagonals. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted. (10 points) Rectangle ABCD is constructed with line EF drawn through its center. If the rectangle is dilated using a scale factor of 3 and a line is drawn through the center of the new dilated figure, what relationship will the new line have with line EF? Explain your reasoning using complete sentences. Matt is constructing two similar triangles for an art project. The drawing below shows Matt's plans, but there is an error in his drawing. What changes would he make to the dimensions to change the error? Explain your reasoning using complete sentences. (10 points) Prove the Pythagorean Theorem using similar triangles. The Pythagorean Theorem states that in a right triangle, the sum of the squares of the lengths of the legs of the triangle equals the squared length of the hypotenuse. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted. (10 A kite is a quadrilateral with two pairs of adjacent, congruent sides. The vertex angles are those angles in between the pairs of congruent sides. Prove the diagonal connecting these vertex angles is perpendicular to the diagonal connecting the nonvertex angles. Be sure to create and name the appropriate geometric figures. This figure does not need to be submitted. (10 points) Stephen is making a map of his neighborhood. He knows the following information: His home, the bus stop, and the grocery store are all on the same street. His home, the park, and his friend's house are on the same street. The angle between the park, the bus stop, and his home is congruent to the angle between his friend's house, the grocery store, and his home. What theorem can Stephen use to determine the remainin Pythagorean Theorem Triangle Proportionality Theorem Midsegment Theorem Side-Angle-Side Similarity Theorem Use ABC shown below to answer the question that follows: What statement is needed to prove that ABC is similar to DBA? (5 points) Segment BC is a hypotenu se. Angle B is congruent to itself. Segment BA is shorter than segment BC. Segment BC is intersecte d by segment AD. Look at the figure. What is the length, in units, of segment CD? (5 points) 5.50 12.2 5 8 14 Triangle MNO is dilated to form new triangle PQR. If angle O is congruent to angle R, what other information will prove that the two triangles are similar? (5 points) Side ON is congrue nt to side RQ. Angle N is congrue nt to angle Q. Side MN is congrue nt to side PQ. Angle M is congrue nt to angle Q. QUESTION 1 Part A: The first incorrect step is step 1 Part B: If they are being picky about wording: Step 1 could be written incorrectly. It does not say to use a compass. It assumes you know where A and B are (rather than telling you to label (as it does in step 4). It does not tell you to use the same compass width to make the two arcs (as it does in step 3 relating to step 2). However, I think those things could be assumed by anyone who knows how to make a perpendicular line through a point. Step 5 contains two errors: you would use a straight edge to connect the points, not the compass. It does not tell you to draw arrows on the line (the problem says to make a line not a line segment) Part C: The compass work for construction of perpendicular line due to te following reasons i. ii. There is accuracy when marking the points for important location such as point A,B and C It is only compass that can draw the arcs and curves. QUESTION 2 The new coordinates for the image are: A (-2, 2) becomes A' (0, -1) B (-2, 4) becomes B' (0, 1) C (2, 4) becomes C' (4, 1) D (2, 2) becomes D' (4, -1) Description of the Characteristics Because all points have been translated in a straight line, in the same direction by the same distance, the new figure A' B' C' D' will have the same shape, same size and same orientation as the original image. It will simply be in a different location on the graph. If you were to join A with A', B with B', and so on, you would see that all 4 vectors are parallel and are of equal length. If you were to compare the segment AB (side AB of the original quadrilateral) with the segment A'B' (in the new quadrilateral), you would see that they are parallel (same slope) and have the same length. Same with the other sides QUESTION 3 Since corresponding sides of these two triangles are equal then the required equation is 5x - 21 = x - 1 And its solution is 5x - x = 21 - 1 4x = 20 x = 20/4 x=5 Since they are congruent, then all the sides are equal. Already, AB=ED=6 and AC=EF=5 so definitely EF=BC QUESTION 4 Proofing that a rectangle has congruent diagonals Since we want to prove "diagonals are congruent,\" Let's assume that "diagonals are not congruent in a rectangle.\" By definition of rectangle, it has four right angles and opposite sides is congruent. So, by Pythagorean Theorem, AB^2 + BC^2 = AC^2 and BC^2 + CD^2 = BD^2. Since opposite sides are congruent, AB=CD, AD = BC. By substitution BC^2 = BD^2. That means diagonals are congruent. . Therefore, the diagonals of rectangle are congruent. QUESTION 5 The relationship new line will have with line EF Both lines are coincident; they both pass through the center of the rectangle. The length of the new line E'F' is 3 times the length of the old line EF. The scale factor multiplies each side length by 3 QUESTION 6 Changes to be made on the dimension to change the error DE should be 6.75 units not 7.5 units The linear scale factor is 1.5 meaning lengths in the second triangle are each 1.5 times the lengths in the first triangle He has two options - decrease DE by 0.75 which is the easiest way to ensure both triangles are similar. In other way, he can alter AC and BC so that they are in the same proportion AB: DE to ensure similarity too. QUESTION 7 Proving the Pythagorean Theorem Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides: QUESTION 8 Proof of diagonal making a kite is perpendicular Given that an image attached with this prove. 1. AD = AB, CD=CB 1. Given 2. OD = OB 2. By definition, the diagonal connecting vertec angles bisects the other diagonal 3. OA = OA 3. By reflexive property 4. Triangle AOB = triangle AOD 4. By SSS congruence 5. Angle AOB = angle AOD 5. By CPCTC 6. Angle AOB = angle AOD = 90 6. both angles add up to 180 7. AC DB 7. Definition of right angles QUESTION 9 Theorem that Stephen should use Side-Angle-Side Similarity Theorem QUESTION 10 Statement needed to prove that ABC is similar to DBA Angle B is congruent to itself QUESTION 11 What is the length, in units, of segment CD? QUESTION 1 Part A: The first incorrect step is step 1 Part B: If they are being picky about wording: Step 1 could be written incorrectly. It does not say to use a compass. It assumes you know where A and B are (rather than telling you to label (as it does in step 4). It does not tell you to use the same compass width to make the two arcs (as it does in step 3 relating to step 2). However, I think those things could be assumed by anyone who knows how to make a perpendicular line through a point. Step 5 contains two errors: you would use a straight edge to connect the points, not the compass. It does not tell you to draw arrows on the line (the problem says to make a line not a line segment) Part C: The compass work for construction of perpendicular line due to te following reasons i. ii. There is accuracy when marking the points for important location such as point A,B and C It is only compass that can draw the arcs and curves. QUESTION 2 The new coordinates for the image are: A (-2, 2) becomes A' (0, -1) B (-2, 4) becomes B' (0, 1) C (2, 4) becomes C' (4, 1) D (2, 2) becomes D' (4, -1) Description of the Characteristics Because all points have been translated in a straight line, in the same direction by the same distance, the new figure A' B' C' D' will have the same shape, same size and same orientation as the original image. It will simply be in a different location on the graph. If you were to join A with A', B with B', and so on, you would see that all 4 vectors are parallel and are of equal length. If you were to compare the segment AB (side AB of the original quadrilateral) with the segment A'B' (in the new quadrilateral), you would see that they are parallel (same slope) and have the same length. Same with the other sides QUESTION 3 Since corresponding sides of these two triangles are equal then the required equation is 5x - 21 = x - 1 And its solution is 5x - x = 21 - 1 4x = 20 x = 20/4 x=5 Since they are congruent, then all the sides are equal. Already, AB=ED=6 and AC=EF=5 so definitely EF=BC QUESTION 4 Proofing that a rectangle has congruent diagonals Since we want to prove "diagonals are congruent,\" Let's assume that "diagonals are not congruent in a rectangle.\" By definition of rectangle, it has four right angles and opposite sides is congruent. So, by Pythagorean Theorem, AB^2 + BC^2 = AC^2 and BC^2 + CD^2 = BD^2. Since opposite sides are congruent, AB=CD, AD = BC. By substitution BC^2 = BD^2. That means diagonals are congruent. . Therefore, the diagonals of rectangle are congruent. QUESTION 5 The relationship new line will have with line EF Both lines are coincident; they both pass through the center of the rectangle. The length of the new line E'F' is 3 times the length of the old line EF. The scale factor multiplies each side length by 3 QUESTION 6 Changes to be made on the dimension to change the error DE should be 6.75 units not 7.5 units The linear scale factor is 1.5 meaning lengths in the second triangle are each 1.5 times the lengths in the first triangle He has two options - decrease DE by 0.75 which is the easiest way to ensure both triangles are similar. In other way, he can alter AC and BC so that they are in the same proportion AB: DE to ensure similarity too. QUESTION 7 Proving the Pythagorean Theorem Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB. Point H divides the length of the hypotenuse c into parts d and e. The new triangle ACH is similar to triangle ABC, because they both have a right angle (by definition of the altitude), and they share the angle at A, meaning that the third angle will be the same in both triangles as well, marked as in the figure. By a similar reasoning, the triangle CBH is also similar to ABC. The proof of similarity of the triangles requires the Triangle postulate: the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the triangles leads to the equality of ratios of corresponding sides: QUESTION 8 Proof of diagonal making a kite is perpendicular Given that an image attached with this prove. 1. AD = AB, CD=CB 1. Given 2. OD = OB 2. By definition, the diagonal connecting vertec angles bisects the other diagonal 3. OA = OA 3. By reflexive property 4. Triangle AOB = triangle AOD 4. By SSS congruence 5. Angle AOB = angle AOD 5. By CPCTC 6. Angle AOB = angle AOD = 90 6. both angles add up to 180 7. AC DB 7. Definition of right angles QUESTION 9 Theorem that Stephen should use Side-Angle-Side Similarity Theorem QUESTION 10 Statement needed to prove that ABC is similar to DBA Angle B is congruent to itself QUESTION 11 What is the length, in units, of segment CD