Review Packet: Ch. 4 & 5 LT13 LT17
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1 Review Packet: Ch. 4 & 5 LT13 LT17 Name: Pd. LT13: I can apply the Triangle Sum Theorem and Exterior angle Theorem to classify triangles and find the measure of their angles. 1. Find x and y. 2. Find x and y. 3. Find x and y. 4. Find the value of x. Classify the triangle by its angles. LT14: I can use properties of midsegments and write coordinate proofs. 1. Find x. Find the length of UV and GI. 2. If the m HUV = 53, what is: a) m HUG b) m GUV The vertices of ABC are A (3,2 ), B(3, 4) and C (1, 6). 1. Find the coordinates of the midsegment of the triangle. 2. Prove that ST = ½ AC and that ST ǁ AC. Triangle NYE has vertices N( 7, 6) Y(2, 7) and E( 3, 2). 1. What kind of triangle is NYE? Show your work. Prove that ABC is isosceles. Given: G and H are midpoints. Prove: GH = ½ DF 1. Find the coordinates of a midsegment in the triangle. 2. Use the slope and distance formula to verify the Midsegment Theorem is true.
2 LT15: I can use properties of perpendicular bisectors and angle bisectors. 1. Find x. 2. Find x. 3. Find x. 4. Find x and FE. LT16: I can use properties of medians and altitudes of triangles. 1. SU is the median. Find x and m SUR. 2. Find the altitude of the isosceles triangle. 3. In the diagram, which special segment is LN? LT17: I can construct the orthocenter, circumcenter, centroid and incenter of a triangle and apply the properties of each to solve real world problems. 1. Which point of concurrency is equidistant from the vertices of a triangle? What is it formed by? 2. Which point of concurrency is equidistant from the sides of a triangle? What is it formed by? 3. Which point of concurrency allows you to construct the largest circle inside a triangle? 4. Which point of concurrency allows you to construct a circle that allows you to inscribe the triangle? 5. The incenter is inside the triangle. (always, sometimes, never) 6. The circumcenter is inside the triangle. 7. Where is the circumcenter of a right triangle? 8. What is the orthocenter formed by? 9. How does the centroid partition the median? 10. Which point of concurrency should you find that is equidistant from three points? Sketch the point of concurrency listed: (label the congruent and perpendicular segments formed with appropriate marks) Incenter Circumcenter Orthocenter Centroid
3 Identify the point of concurrency in each diagram. Find the circumcenter of RSO with vertices R( 6, 0), S (0, 4), and O (0, 0) by finding the perpendicular bisectors of each side. 1. Find the coordinates of the midpoint of HG and call it point M. 2. Draw the median from vertex H. 3. Find the coordinates of the centroid, call it point P. 4. Prove that IP = 2 3 IM Find x. L is the centroid. Find x if: ML = 10x 4 and MR = 12x + 18 You want to place a decoration on the centroid of the triangle. How far down from point A should you place the decoration? G is the incenter. Find the length of GD.
4 A committee has decided to build a park in Deer County. The committee agreed that the park should be equidistant from the three largest cities in the county, which are labeled X, Y, and Z in the diagram. Explain why this may not be the best place to build the park. Use a sketch to support your answer. Use the right triangle below. The circumcenter of a right triangle is always the midpoint of the hypotenuse. 1) Prove that the midpoint of the hypotenuse of a right triangle is equidistant from all three vertices (circumcenter theorem). Special Segment Point of Concurrency Sketch Properties Midsegment Perpendicular Bisector Angle Bisector Median Altitude D = (x x ) 2 + (y y ) 2 m = y 2 y y1 y M ( xm, ym), x 2 x y, y y3
5
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