Date: Period: Directions: Answer the following questions completely. Please remember to show all work that is necessary for the test.

Transcription

1 Name: Similar Triangles Review Sheet Date: Period: Geometry Honors Directions: Answer the following questions completely. Please remember to show all work that is necessary for the test. Ratio of Similitude: 1) The sides of a triangle are 5, 6, and 10. Find the longest side of the triangle, if the shortest side is 15. 2) Two triangles are similar. The lengths of the sides of the smaller triangle are 4, 6, 7. The shortest side of the larger triangle is 16. What is the length of the largest side? 3) A boy looks into a mirror that has been placed on the ground 3 meters away and sees the reflection of the top of a telephone pole. If the mirror is 21 meters away from the telephone pole and the person is 2 meters tall, how tall is the telephone pole? 4) The sides of a quadrilateral measure 12, 15, 24, and 18. If the shortest side of a similar quadrilateral measures 4, find the measures of the remaining sides of this quadrilateral. 5) In the accompanying diagram, is similar to and. What is the length of 6) and inches,, and. What is the length of DE? (a) 8.3 inches (b) inches (c) inches (d) 39 inches 7) Henry casts a shadow, 3 feet in length. His son who is 3.5 feet tall, casts a shadow that is 1.8 feet in length. Which of the following best represents Henry s height? (a) 5 ft 6 in (c) 5 ft 8 in (c) 5 ft 10 in (d) 6 ft 8). Find the perimeter of

2 Overlapping Triangles: 1) Find the value of x. 2) Find the length of FJ. 3) Find the value of n. 4) Find the value of x. 5) Find the value of the height (h) in the following diagram at which the tennis ball must be hit so that it will just pass over the net and land 6 meters away from the base of the net. Midsegment: 1) In the diagram below, the vertices of ΔDEF are the midpoints of the sides of equilateral triangle ABC, and the perimeter ΔABC is 36 cm. What is the length, in centimeters of EF? (1) 6 (2) 12 (3) 18 (4) 4 2) In the diagram below of ΔACT, D is the midpoint of AC, O is the midpoint of AT, and G is the midpoint of CT. If AC = 10, AT = 18, and CT = 22, what is the perimeter of parallelogram CDOG? (1) 21 (2) 25 (3) 32 (4) 40

3 3) MT is a midsegment of. Use the diagram to the right to answer the following questions. (a) What is the measure of? (b) What is the measure of? (c) What is the measure of (d) What is the measure of? 4) A student makes the rabbit shadow that you see in the diagram. The part of his hand making the shadow is 4.5 inches tall, and it is halfway between the flashlight and the wall. The top of his fingers are halfway between the flashlight and the top of the shadow. (a) How tall is the rabbit? (b) Now assume that the boy s hand is 3.7 inches and the shadow is 7.6 inches. Is the part of the boy that creates the shadow a midsegment? 5) X, Y, and Z are midpoints of the sides of triangle ABC. XY is, and. How long is 6) Given construct midsegment XY, if the midpoint of AB is X and the midpoint of AC is Y. [Leave all construction marks.] Ratio of Similarity: 1) If the ratio of the surface area of two similar cylinders is 16:25, what is the ratio of their volumes? 2) If the ratio of the perimeters of two similar triangles is 2:9, what is the ratio of their areas? 3) If the ratio of the volumes of two similar spheres is 125:8, what is the ratio of their areas?

4 4) If the ratio of the surface areas of two similar rectangular boxes is 4:9, what is the volume of the larger box if the volume of the smaller box is 26 m 3? (round to the nearest tenth, if necessary) 5) If the ratio of the volumes of two cylinders is 8 : 343, what is the area of the smaller cylinder if the area of the larger cylinder is 98 m 2? 6) If the ratio of the perimeters of two similar spheres is 4 : 5, what is the volume of the larger sphere if the volume of the smaller sphere is 75 in 3? If necessary, round your answer to the nearest hundredth. Not-So-Formal Proofs: 1) has angles with measures of 60 degrees and 80 degrees. has angles with measures of 80 degrees and 50 degrees. Are these triangles similar? 2) Show that the two triangles are similar. 3) Determine if the right triangles below are similar. Justify your answer. 4) Is? Explain. 5) Is? Explain. 6) Which of the followng side lengths coould represent two similar triangles? (a) 12 in, 8 in, 6 in : 24 in, 16 in, 10 in (b) 9 in, 4 in, 8 in : 13.5 in, 6 in, 12 in (c) 5 in, 7 in, 11 in : 12.5 in, 14 in, 27.5 in (d) 7 in, 12in, 9 in : 5.25 in, 9 in, 4.5 in Similar Triangle Proofs: 1) Given: 2) Given: is isosceles with base AC

5 3) Given: 4) Given: 5) The diagram below shows, with and. Prove that is similar to

6 Name: Review Sheet Date: Period: Geometry Honors Directions: Answer the following questions completely. Please remember to show all work that is necessary for the test. 1) Given: 2) In a triangle, the measure of the first angle is less than the measure of the second one, and the measure of the third angle is twice as great as the measure of the first one. What are the measures of the angles of the triangle? 3) Farmington, NY, has plans for a new triangular park. If plotted on a coordinate grid, the vertices would be A(3,3), B(5,-2), and C(-3,-1). However, a tract of land has become available that would enable the planners to increase the size of the park, which is based on the following transformation of the original triangular park,. On the grid to the right, graph and label both the original park and its image, the new park, following the transformation. 4) Dorothy says that two angles in a triangle she drew 5) Given,, have the measures and. Can Dorothy be right, and, what is about her triangle? Explain why or why not.? 6) For question number 6, use the diagram to answer the questions that follow. (a) What is the length of? (b) What is the length of? (c) What is the length of? (d) What is the perimeter of?

7 7) In the diagram below, bisects angle ADF. Use the 8) Find the equation of a line that is perpendicular angle measures given on the diagram to find the angles to and passes through the point of triangle ADE. (-4,8). 9) The diagram shows two moves of a checkers piece. 10) Using a compass and a straightedge, construct an Describe its movement as a translation. Use the row and angle bisector. [Leave all construction marks.] Column numbers as coordinates. 11) On the ray drawn below, using a compass and a straightedge, construct an equilateral triangle with a vertex at R. The length of a side of the triangle must be equal to a length of the diagonal of rectangle ABCD. [Leave all construction marks.]

8 Answer Key: Ratio of Similitude: 1.) 30 2.) 28 3.) 14 m 4.) 5, 8, 6 5.) 10 6.) (c) 7.) (c) 8.) 9.7 Overlapping Triangles: 1.) 40 2.) x = 4; FJ = 10 3.) ) 10 5.) 2.7 Midsegment: 1.) (1) 2.) (3) 3.) (a) 79 o (b) 67 o (c) 34 o (d) 146 o 4.) (a) 9 inches (b) No because his hand is not half the length of the rabbit shadow 5.) k = 9; XY = 26 6.) correct construction Ratio of Similarity: 1.) 2.) 3.) 4.) 87.8 m 3 5.) 8m 2 6.) in 3 Not-So-Formal Proofs: 1.) No, not all of the angle measures are congruent. 2.) SAS~ 3.) SAS~ 4.) SSS~ 5.) Not similar because not all of the sides are in proportion. 6.) (b) Similar Triangle Proofs (Formal): 1.) Statements: Reasons: Given Intersection lines form congruent vertical angles. 2.) Statements: Reasons: is isosceles with base AC Given Base angles of an isosceles triangle are congruent. Perpendicular lines meet to form congruent right angles. Corresponding sides of similar triangle are in proportion. 3.) Statements: Reasons: Given, Parallel lines cut by a transversal form congruent alternate interior angles. Corresponding sides of similar triangle are in proportion. The product of the means equals the product of the extremes.

9 4.) Statements: Reasons: ; Given Perpendicular lines form congruent right angles. Corresponding sides of similar triangle are in proportion. 5.) Statements: Reasons: ; Given Reflexive Review Topics The product of the means equals the product of the extremes. 1.) Statements Reasons BD is an altitude of Given BD is a median of An altitude creates congruent right angles. A median divides a segment into two congruent segments. Reflexive SAS 2.) 54 o, 42 o, 84 o 3.) A (6,-6), B (-4,-10), C (-2,6) 4.) No, there is only 180 o in a triangle. 5.) x = 36; m<8=74 o 6.) (a) 11 (b) 11 (c) 13 (d) 34 7.) m<ead = 48 o, m<ade = 66 o, m<dea = 66 o 8.) 9.) 10.) correct construction 11.) correct construction