Similar Polygons

Two polygons are similar if and only if their corresponding angles are congruent and the measures of the corresponding sides are proportional.

is read is similar to

The order of the vertices in similarity statement is the same as a congruency statement.

ABCDEFGH

A B

CD

E

GH

F

Rectangle WXYZ is similar to rectangle PQRS with a scale factor of 1.5. If the length and width of rectangle PQRS are 10 meters and 4 meters, respectively, what are the length and width of rectangle WXYZ?

Write proportions for finding side measures. Let one long side of each WXYZ and PQRS be and one short side of each WXYZ and PQRS be

Answer:

Quadrilateral GCDE is similar to quadrilateral JKLM

with a scale factor of If two of the sides of GCDE

measure 7 inches and 14 inches, what are the

lengths of the corresponding sides of JKLM?

Answer: 5 in., 10 in.

The scale on the map of a city is inch equals 2

miles. On the map, the width of the city at its widest

point is inches. The city hosts a bicycle race

across town at its widest point. Tashawna bikes at

10 miles per hour. How long will it take her to

complete the race?

Explore Every equals 2 miles. The

distance across the city at its widest point is

Solve

Cross products

Divide each side by 0.25.

The distance across the city is 30 miles.

Plan Create a proportion relating the measurements to the scale to find the distance in miles. Then use the formula to find the time.

Divide each side by 10.

Answer: 3 hours

It would take Tashawna 3 hours to bike across town.

Examine To determine whether the answer is reasonable, reexamine the scale. If 0.25 inches 2 miles, then 4 inches 32 miles. The distance across the city is approximately 32 miles. At 10 miles per hour, the ride would take about 3 hours. The answer is reasonable.

An historic train ride is planned between two landmarks on the Lewis and Clark Trail. The scale on a map that includes the two landmarks is 3 centimeters = 125 miles. The distance between the two landmarks on the map is 1.5 centimeters. If the train travels at an average rate of 50 miles per hour, how long will the trip between the landmarks take?

Answer: 1.25 hours

Triangle Proportionality Theorem

If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into segments of proportional lengths.

Example

From the Triangle Proportionality Theorem,

In and Find SU.

S

Answer: 15.75

In and Find BY.

B

Converse of the Triangle Proportionality Theorem

If a line intersect two sides of a triangle and separates the sides into corresponding segments of proportional lengths, the line is parallel to the third side.

Example

In and Determine

whether Explain.

In and AZ = 32.

Determine whether Explain.

Answer: No; the segments are not in proportion since

X

Triangle Midsegment Theorem

Midsegment of a triangle – a segment whose endpoints are the midpoints of two sides of the triangle

Triangle Midsegments Theorem – A midsegmet of a triangle is parallel to one side of the triangle, and its length is one-half the length of that side.

Example

Triangle ABC has vertices A(–2, 2), B(2, 4,) and C(4, –4). is a midsegment of Find the coordinates ofD and E.

(-2, 2)

(2, 4)

(4, –4)

Triangle ABC has vertices A(–2, 2), B(2, 4) and C(4, –4).

is a midsegment of Verify that

(-2, 2)

(2, 4)

(4, –4)

Triangle ABC has vertices A(–2, 2), B(2, 4) and C(4, –4).

is a midsegment of Verify that

(-2, 2)

(2, 4)

(4, –4)

Triangle UXY has vertices U(–3, 1), X(3, 3), and Y(5, –7). is a midsegment of

a. Find the coordinates of W and Z.

b. Verify that

c. Verify that

Answer: W(0, 2), Z(1, –3)

Answer: Since the slope of and the slope of

Answer: Therefore,

Proportional Perimeters Theorem

If two triangles are similar, then the perimeters are proportianal to the measures of corresponding sides.

If and find the perimeter of

Let x represent the perimeter of The perimeter of

C

If and RX = 20, find the perimeter of

Answer:

R

Special Segments of Similar Triangles

If two triangles are similar, then the measures of the corresponding altitudes are proportional to the measures of the corresponding sides.

Example

If two triangles are similar, then the measures of the corresponding angle bisectors of the triangle are proportional to the measures of the corresponding sides.

Example

If two triangles are similar, then the measures of the corresponding medians are proportional to the measures of the corresponding sides.

Example

In the figure, is an altitude of and is an altitude of Find x ifand

K

Answer: 17.5

N

In the figure, is an altitude of and is an altitude of Find x if and

The drawing below illustrates the legs, of a table. The top of the legs are fastened so that AC measures 12 inches while the bottom of the legs open such that GE measures 36 inches. If BD measures 7 inches, what is the height h of the table?

Answer: 28 in.