Name ______________________________
Ratio and Similarity
Module 2
Learning Target: I can apply the concept that if two triangles are similar, corresponding pairs of
angles have the same measure and corresponding sides are proportional.
Opening Exercises
1. If ∆QRS  ∆ZYX, identify the pairs of congruent angles and the pairs of congruent sides.
2. Solve the proportion.
Ratio and Similarity
A ratio compares two numbers by division. The ratio of two numbers a and b can be written as a to b,
𝑎
1
a:b, or 𝑏 , where b ≠ 0. For example, the ratios 1 to 2, 1:2, and 2 all represent the same comparison.
1. The ratio of the side lengths of a triangle is 4:7:5, and its perimeter is 96 cm. What is the length of
the shortest side?
2. The ratio of the angle measures in a triangle is 1:6:13. What is the measure of each angle?
Figures that are similar (~) have the same shape but not necessarily the same size. If two triangles
are similar, corresponding pairs of angles have the same measure and corresponding sides are
proportional.
3. Identify the pairs of congruent angles and corresponding sides.
A similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons. The
3
1
6
similarity ratio of ∆ABC to ∆DEF is 6 , or 2 . The similarity ratio of ∆DEF to ∆ABC is 3 , or 2.
4. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity
statement.
5. The accompanying diagram shows two similar triangles. Solve for x.
Name ______________________________
Ratio and Similarity
Module 2
Problem Set
1. A triangle has sides whose lengths are 5, 12, and 13. A similar triangle could have sides with
lengths of
1)
2)
3)
4)
3, 4, and 5
6, 8, and 10
7, 24, and 25
10, 24, and 26
2. In the accompanying diagram, ∆𝑄𝑅𝑆 is similar to ∆𝐿𝑀𝑁, 𝑅𝑄 = 30, 𝑄𝑆 = 21, 𝑆𝑅 = 27, and 𝐿𝑁 =
7. What is the length of ̅̅̅̅
𝑀𝐿?
3. The Rivera family bought a new tent for camping. Their old tent had equal sides of 10 feet and a
floor width of 15 feet, as shown in the accompanying diagram. If the new tent is similar in shape to
the old tent and has equal sides of 16 feet, how wide is the floor of the new tent?
4. The accompanying diagram shows a section of the city of Tacoma. High Road, State Street, and
Main Street are parallel and 5 miles apart. Ridge Road is perpendicular to the three parallel streets.
The distance between the intersection of Ridge Road and State Street and where the railroad tracks
cross State Street is 12 miles. What is the distance between the intersection of Ridge Road and Main
Street and where the railroad tracks cross Main Street?
5. As shown in the diagram below, ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹, 𝐴𝐵 = 7𝑥, 𝐵𝐶 = 4, 𝐷𝐸 = 7, and 𝐸𝐹 = 𝑥. What is
the length of ̅̅̅̅
𝐴𝐵 ?
6. In the diagram below, ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹, 𝐷𝐸 = 4, 𝐴𝐵 = 𝑥, 𝐴𝐶 = 𝑥 + 2, and 𝐷𝐹 = 𝑥 + 6. Determine
the length of ̅̅̅̅
𝐴𝐵 . [Only an algebraic solution can receive full credit.]
7. If ∆𝐴𝐵𝐶~∆𝑍𝑋𝑌, 𝑚∠𝐴 = 50, and 𝑚∠𝐶 = 30, what is 𝑚∠𝑋?
8. In the diagram below, ∆𝐴𝐵𝐶~∆𝐸𝐹𝐺, 𝑚∠𝐶 = 4𝑥 + 30, and 𝑚∠𝐺 = 5𝑥 + 10. Determine the
value of x.
9. The base of an isosceles triangle is 5 and its perimeter is 11. The base of a similar isosceles triangle
is 10. What is the perimeter of the larger triangle?
10. Which is not a property of all similar triangles?
1) The corresponding angles are congruent.
2) The corresponding sides are congruent.
3) The perimeters are in the same ratio as the
corresponding sides.
4) The altitudes are in the same ratio as the
corresponding sides.
Name ______________________________
Ratio and Similarity
Module 2
Exit Ticket
1. Two triangles, ∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹, are in the plane so that ∠𝐴 = ∠𝐷, ∠𝐵 = ∠𝐸, ∠𝐶 = ∠𝐹, and
𝐷𝐸
𝐸𝐹
𝐷𝐹
=
=
. Explain why the triangles must be similar.
𝐴𝐵
𝐵𝐶
𝐴𝐶
2. On a scale drawing of a new school playground, a triangular area has sides with lengths of 8
centimeters, 15 centimeters, and 17 centimeters. If the triangular area located on the playground
has a perimeter of 120 meters, what is the length of its longest side?