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Right Angles and Ratios
Slideshow 40, Mathematics Mr. Richard Sasaki
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Objectives Understand common types of right-angled triangles
See how ratios relate for given angle sizes Be able to show that a triangle is right-angled Be able to show whether other triangles are acute or obtuse
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Right-Angled Scalene Triangles
A right-angled triangle could be or isosceles scalene A scalene triangle is a triangle where no lengths or angles are equal. A commonly known scalene triangle is the triangle. Here, the angles are in ratio 1:2:3 and the edges are in the ratio 1: 3 :2. 30 π The purple triangle is also and edges are in the same ratio. 2 5 3 3 5 60 π 60 π 30 π 1 10
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Right-Angled Isosceles Triangles
The only type of isosceles right-angled triangle is the triangle. An isosceles triangle is a triangle where a pair of lengths and angles are equal. Here, the angles are in ratio 1:1:2 and the edges are in ratio 1:1: 2 . 45 π 2 The purple triangle is also and edges are in the same ratio. 1 4 2 4 45 π 4 1
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7 2 ππ 6 45 π 2 3 30 π 60 π 3 2 30 π 45 π 45 π 3 2 2 3 60 π 6 3 3 ππ 3 2π π 3 π ππ 3 π ππ ππ π΄πππ: πβ 2 = π π 2 2.5 2.5
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Testing if Right-Angled
We know that for a right-angled triangle, π 2 + π 2 = π 2 where π and π are legs and π is the hypotenuse. We can use the opposite principle. If π 2 + π 2 = π 2 is satisfied, the triangle must be right-angled. Example Consider a triangle with edges 15 ππ, 36 ππ and 39 ππ. Show that it is right-angled. As π=15, π=36 and π=39, π 2 + π 2 = π 2 must be satisfied = This is simplified to make =1521 which equates. β΄ the triangle is right-angled.
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Yes No No No Yes Yes Two of the numbers are even, so it is not right-angled. As π=39, π=80 and π=89, π 2 + π 2 = π 2 must be satisfied = This is simplified to make =7921 which equates. β΄ the triangle is right-angled. As π=51, π=81 and π=120, π 2 + π 2 = π 2 must be satisfied = This is simplified to make β 14400 which as shown does not equate. β΄ the triangle is not right-angled.
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Right-angled Isosceles
Acute and Obtuse Triangles An acute triangle exists when all angles are less than 90 π . An obtuse triangle exists when one angle is greater than 90 π A right-angled triangle exists when one angle is exactly 90 π . Acute Isosceles Obtuse Isosceles Right-angled Isosceles Acute Scalene Obtuse Scalene Right-angled Scalene
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Acute and Obtuse Triangles
We know that to be right-angled, π 2 + π 2 = π 2 must be satisfied. But how do we show if a triangle is acute or obtuse? When a triangle is acuteβ¦ π 2 + π π 2 > When a triangle is obtuseβ¦ π 2 + π π 2 < Note: Here there is often no hypotenuse. Allow π to represent the longest length or one of the longest lengths.
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Acute and Obtuse Triangles
Example A triangle has edges 4 ππ, 7 ππ and 9 ππ. State what type of triangle it is. π 2 + π 2 ___ π 2 ___ 9 2 16+49___81 65 ___81 < As π 2 + π 2 < π 2 , the triangle must be obtuse All edges are different so the triangle is obtuse scalene.
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(disputed) Obtuse Isosceles Acute Scalene Right-Angled Scalene Acute Scalene Obtuse Scalene Acute Isosceles Right-Angled Scalene Obtuse Scalene Obtuse Scalene The length of the two shorter edges is less than the longest edge.
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Obtuse angles are over 90 π so a triangle with two would be over 180 π .
π+π>π All angles are nearly the same. One angle is almost 180 π and the other two are very small. One angle is small (opposite the base). Other two are nearly 90 π . Two lengths would be huge in comparison to the third unless one angle is close to 180 π .
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