Proportions, Similarities

Question 1

What is a comparison of two (or more) quantities?

Answer 1

a ratio

a:b, a to b

Question 2

The sides of a pentagon are in the ratio 2:3:5:1:4. If the perimeter of the pentagon is 90 units, find the lengths of the five sides.

Answer 2

x = 6

12, 18, 30, 6, and 24 units

Question 3

Is ∆ABC similar to ∆DBE?

Answer 3

Yes

Since <A is congruent to <BDE, and <B is shared by both triangles, we have the similarity of the triangles by AA.

Question 4

Find BE.

Answer 4

7.5

Question 5

Find DE.

Answer 5

6 = x

Question 6

Are these triangles similar?

Answer 6

Yes

Question 7

Triangle PQR is similar to triangle FGH.

Solve for n.

Answer 7

n = 6

Question 8

Triangle XYZ is similar to triangle ABC.

Solve for k.

Answer 8

k = 10.5

Question 9

Triangle FGH is similar to triangle DEF.

Solve for p.

Answer 9

p = 16

Similar triangles have proportional sides. Therefore, we can set up equivalent proportions and solve for p.

Question 10

OL is perpendicular to DX.

Find the length of OX.

Answer 10

7.09

Strategy

Denote OX as x. Then, OD = 13 - x.

We can relate OX and OD using a proportion from similar triangles and solve the resulting equation to find x.

∆POD is similar to ∆BOX by angle-angle similarity:
* <X = <D since they are both right angles.
* <LOD and <LOX are right angles because OL is perpendicular to DX.
* <1 = <4 since they are complementary to congruent angles.

Setting up a proportion

Corresponding sides of similar triangles are proportional, so we can set up and solve a proportion:

OX OB

OD OP

We picked side OX because we need to find its length. We picked side OD because it corresponds to side OX. Then we picked another pair of corresponding sides where we already kne wthe measurements.