48 TRIANGLES

Formula

Let n = number of rows

If n is even:

Total no. of ∆𝑠 = 𝑛(𝑛+2)(2𝑛+1)

8

If n is odd:

Total no. of ∆𝑠 = 𝑛+1 (2𝑛2+3𝑛 −1)

8

n = 5

Total no. of ∆𝑠 = 𝑛+1 (2𝑛2+3𝑛 −1)

8

Total no. of ∆s =5+1 [2(5)2+3 5 −1]

8

= 6 (50+14)

8

= 6(64)

8

= 384

8

= 48

SIMILARITIES IN

RIGHT TRIANGLES

Theorem 1

The altitude to the hypotenuse

of a right triangle separates

the right triangle into two

triangles which are similar to

each other and to the original

triangle.

Example

In a right ∆𝐴𝐵𝐶, 𝐵𝐸 is an altitude.

Three similar triangles:

∆𝑨𝑩𝑪 ~ ∆𝑨𝑬𝑩 ~ ∆𝑩𝑬𝑪

Congruent angles:

BEC ≅ AEB ≅ ABC

A ≅ EBC

ABE ≅ ECB

Theorem 2In any right triangle,

a.The altitude to the hypotenuse is the

geometric mean between the

segments into which it separates the

hypotenuse.

b.Each leg is a geometric mean of the

hypotenuse and the segment of the

hypotenuse adjacent to the leg.

B

A E C

Three pairs of similar triangles:

∆AEB ~ ∆BEC

∆BEC ~ ∆ABC

∆AEB ~ ∆ABC

Proportions:

a.∆AEB ~ ∆BEC, 𝑨𝑬

𝑩𝑬= 𝑩𝑬

𝑪𝑬

b.∆BEC ~ ∆ABC, 𝑨𝑪

𝑪𝑩= 𝑪𝑩

𝑪𝑬

c.∆AEB ~ ∆ABC, 𝑨𝑪

𝑨𝑩= 𝑨𝑩

𝑨𝑬

𝑆𝑖𝑑𝑒

𝐴𝑙𝑡𝑖𝑡𝑢𝑑𝑒= 𝐴𝑙𝑡𝑖𝑡𝑢𝑑𝑒

𝑆𝑖𝑑𝑒

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝐿𝑒𝑔=

𝐿𝑒𝑔

𝑆𝑖𝑑𝑒

Example 1

Given: s1 = 3, s2 = 9

Unknown: altitude h

Example 2

Solve for side x of big∆.

Example 3

Solve for side x of small∆.

Exercises

Right ∆RAE, with 𝐴𝑃 an altitude.

RP

A E

1.If RP = 3 and PE = 8, find AP and AR.2.If AP = 8 and PE = 12, find RE and AE.

Generalization

Two triangles are similar if their corresponding angles are

congruent.

The altitude to the hypotenuse of a right triangle separates

the right triangle into two triangles which are similar to each

other and to the original triangle.

In any right triangle, the altitude to the hypotenuse is the

geometric mean between the segments into which it

separates the hypotenuse, and each leg is a geometric mean

of the hypotenuse and the segment of the hypotenuse

adjacent to the leg.

AssignmentFinding the Height of a

Roof:

A roof has a cross section

that is a right angle. The

diagram shows the

approximate dimensions

of this cross section.

A. Identify the similar

triangles.

B. Find the height h of

the roof.

THE END