Unit 7A
By: Shrey and Champa
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Overview of Key Concepts
Radicals Review
● Radicals - A quantity expressed as the root of another quantity
● If the altitude is drawn to the hypotenuse of a right triangle, then the
two triangles formed are SIMILAR to the original triangle and to each
other
Altitude & Leg Rule
● Altitude rule : part of hypotenuse =
altitude
altitude
other part of hypotenuse
● Leg Rule : hypotenuse =
leg
leg
projection of leg
2
Pythagorean Theorem
● a²+b²= c²
1st leg
hypotenuse
2nd leg
Pythagorean Triples
● Pythagorean triples - A set of three positive numbers that can fit the
equation c² =a²+b²
● Common Triples are :
3,4,5
5,12,13
8,15,17
7,24,25
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45°- 45°- 90°
● ONLY in 45°-45°-90° triangles, the hypotenuse is √2 times the length of
each leg
x
x√2
x
30°- 60°- 90°
● ONLY in 30°-60°-90° triangles, the hypotenuse is twice as long as the
projection of the 30°. The projection of the 60° is √3 times as much as
the projection of the 30°. The 30° leg is half the length as the
hypotenuse
2X
30°
60 °
X√3
X
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Converse of the Pythagorean Theorem
● If the length of both legs in a triangle are added, and they equal the
length of the hypotenuse, then the triangle is a right triangle
c² = a²+b²
Ex:
1)
(3√13)² = 6²+9²
6
3√13
9*13=36+81
117=117
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2)
80² = 45²+79²
45
80
6400 = 2025+6241
6400 8266
79
5
● If c² < a²+b² ,then triangle ABC is acute
● If c² > a²+b² , then triangle ABC is obtuse
Pythagoras in 3-D
Find the diagonal of a rectangular prism with a length of 8, width of 9 and a
height of 10.
d= √L²+W²+H²
d= √8²+9²+10²
d= √64+81+100
d= √245
d= 7√5
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Common Mistakes and Struggles
❖ Radical Review: Keeping the square root(√) in the denominator, not
simplifying the square root all the way.
❖ Using similar right triangles: Using the wrong ratio or proportion
❖ Using similar right triangles(With Algebra): Forgetting the quadratic formula
❖ Pythagorean Theorem: a²+b²=c² is the formula, people forget to find the
square root of the c², and they find the wrong length of the hypotenuse .
❖ Pythagorean Triples and Reduction:People may be confused and mix up
the triples. For example, people may mix up 8-15-17 and 5-12-13 right
triangles.
❖ Converse of the Pythagorean Theorem: People think that a²+b²<c² means
that the triangle is acute, and they think a²+b²>c² means that the triangle is
obtuse. Actually, it’s the other way around, a²+b²<c² means that the
triangle is obtuse and a²+b²>c² means that the triangle is acute.
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Common Mistakes and Struggles(Part 2)
❖ Pythagorean Theorem in 3-D shapes: In a 3-D solid, one may use the
wrong diagonal in the solid. In a pyramid, they could use the slant height
instead of the edge when they are using the Pythagorean Theorem.
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Connections to Unit 11
In this unit, there is Pythagorean Theorem in 3-D
shapes. In Unit 11, if you only know the altitude and one
side of the base, then you have to use the Pythagorean
Theorem to find the slant height. Also, for the cone, you
have to use the Pythagorean Theorem to find the altitude if
you only know the slant height and the base length.
Additionally, if you only know the slant height and the base
length, you have to use the Pythagorean Theorem to find
the altitude. Lastly, if you only know the altitude and the
slant height, then you have to use the Pythagorean
Theorem to find the base length.
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EASY
**not drawn to scale**
C
10
B
A
D
2
2 = CD
CD DB
2 = 10
10
X
100 = 2X
50 = X
10
MEDIUM
drawn to scale**
8
**not
6
X
3.6
This whole line is 10 units
6=
Y
10 6
10Y= 36
Y=3.6
X=3.6
6.4 X
X²= 23.04
X = 4.8
11
HARD
**not drawn to
scale**
Find the altitude of an isosceles trapezoid with the sides of 17,17,10 and 26:
10
X
17
17
X²+8² = 17²
X²+64 = 289
X = √225
X = 15
26
8
8
12
Real Life Situation
There is a ladder that is leaning on a house that is 30 feet
tall, and the ladder is 12.5 feet away from the house. The
owners of the house are trying to find out the length of the
ladder.
House
Ladder
12.5 ft
30 ft
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Solution to Real Life Situation
❖ Solution 1: a² +b²=c²,
30² +12.5²=c²,
900+156.25=c²,
1056.25=c²,
c=32.5 feet, which means the ladder is 32.5 feet long.
❖ Solution 2: This is a 5-12-13 triangle which is as pythagorean triple.
5*(2.5)=12.5, and 12*(2.5)=30. In this case, you would have to multiply 13
by 2.5 to get the length of the ladder. The length of the ladder is 32.5 feet.
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