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Mathematics 4. Teaching Demo
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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