<p>Name: Date: Page 1 of 5</p><p>Activity 2.3.7 Altitudes and Medians</p><p>Two special segments of a triangle are the medians and the altitudes. This activity will help you to discover where the altitudes and medians are located in a triangle and how to construct them. Coordinate geometry will be used to show the relationship between altitudes and medians in different types of triangles.</p><p>Median (of a triangle) – A segment whose endpoints are a vertex and the midpoint of the side opposite the vertex.</p><p>Altitude (of a triangle) – A perpendicular segment from a vertex to the line containing the side opposite the vertex.</p><p>1. Plot the vertices of and connect them to form a triangle.</p><p> a. What type of triangle did you create, isosceles or scalene? ______b. Find the coordinate of the midpoint of side 10 of the triangle. ( ___ , ___ ). 8 Name that point M. 6 Now connect point B to point M with a straightedge. 4 is a(n) ______of . 2</p><p>Now use your straightedge and create a segment -10 -8 -6 -4 -2 2 4 6 8 10 from point B that is perpendicular to -2 -4 d. The coordinates of the point where the segment -6 intersects this line are ( ___ , ___ ). -8 Name this point L. -10 e. is a(n) ______of .</p><p>What do you notice about the median and the altitude of :</p><p> f. Which is longer?</p><p> g. Are they in the same position?</p><p> h. Are they inside or outside of the triangle?</p><p>Activity 2.3.7 Connecticut Core Geometry Curriculum Version 3.0 22</p><p>2. Plot the vertices of and connect them to form a triangle. </p><p> a. What type of triangle did you create, isosceles or scalene? ______b. Find the coordinate of the midpoint of side 10 of the triangle. ( ___ , ___ ). 8 Name that point M. 6 Now connect point P to point M with a straightedge. 4 2 is a(n) ______of . -10 -8 -6 -4 -2 2 4 6 8 10 Now use your straightedge and create a segment -2 from point P that is perpendicular to . -4 d. The coordinates of the point where the segment -6 intersects this line are ( ___ , ___ ). -8 Name this point L. -10 e. is a(n) ______of .</p><p>What do you notice about the median and the altitude of :</p><p> f. Which is longer?</p><p> g. Are they in the same position?</p><p> h. Are they inside or outside of the triangle?</p><p>2 Name: Date: Page 3 of 5</p><p>3. Plot the vertices of and connect them to form a triangle. </p><p> a. What type of triangle did you create, isosceles or scalene? ______10</p><p>Recall the midpoint formula 8 Midpoint = 6 b. Use this formula to find the midpoint of : 4 2 (_____ , _____ ) Name this point M. -10 -8 -6 -4 -2 2 4 6 8 10 Now connect point T to point M with a straightedge. -2 -4 c. is a(n) ______of . -6 In order to find a perpendicular line from T to , -8 we must find the slope of . -10 d. Slope of = ______</p><p>Recall that slopes of perpendicular lines are opposite reciprocals of each other. e. The opposite reciprocal of slope = ______</p><p>Now start at point T using rise and run and the opposite reciprocal slope to find a point that is on a line perpendicular to . </p><p>Using your straightedge, connect the two points to create a segment from point T that is perpendicular to . Name the point L, where this perpendicular line intersects. f. What are the coordinates of L? ______g. is a(n) ______of .</p><p>What do you notice about the median and the altitude of ?</p><p> h. Which is longer?</p><p> i. Are they in the same position?</p><p> j. Are they inside or outside of the triangle?</p><p>4. Plot the vertices of and connect them to form a triangle. </p><p>Activity 2.3.7 Connecticut Core Geometry Curriculum Version 3.0 44</p><p> a. What type of triangle did you create, isosceles or scalene? ______</p><p>10 b. Find the coordinate of the midpoint of side 8 of the triangle. (___, ___). 6 4 Name that point M. 2 Now connect point D to point M with a straightedge. -10 -8 -6 -4 -2 2 4 6 8 10 c. is a(n) ______of . -2 -4 Now use your straightedge and create a segment -6 from point D that is perpendicular to the -8 line containing . -10 d. The coordinates of the point where the segment intersects this line are ( ___ , ___ ). Name this point L e. is a(n) ______of .</p><p>What do you notice about the median and the altitude to side of :</p><p> f. Which is longer?</p><p> g. Are they in the same position?</p><p> h. Are they inside or outside of the triangle?</p><p>5. Plot the vertices of and connect them to form a triangle. 4 Name: Date: Page 5 of 5</p><p> a. How can you tell that this is an isosceles triangle? 10 b. Which side is the base? 8 c. Which angle is the vertex angle? 6 d. Find the coordinate of the midpoint of the base 4 of the triangle. (___, ___ ). 2</p><p>Name that point M. -10 -8 -6 -4 -2 2 4 6 8 10 Now connect point G to point M with a straightedge. -2 e. is a(n) ______of . -4 -6 f. Is perpendicular to the base? ______-8 g. GM is also an ______of -10 h. Use the midpoint formula to find the midpoint of : (___,___). Call that point N. i. is called a ______of the isosceles triangle. j. Draw segment . is a median of . Is it also an altitude?_____ Explain</p><p>6. Conclusions a. After completing the activity, we can see that in scalene triangles, the ______and ______are distinct line segments. b. In ______triangles, the altitude and median drawn from the vertex angle to the base coincide. c. A(n) ______of a triangle is always inside the triangle, but a(n) ______may be inside or outside. d. In a scalene triangle, when an altitude and a median are drawn from the same vertex, which one is longer? ______</p><p>Activity 2.3.7 Connecticut Core Geometry Curriculum Version 3.0</p>