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Understanding Triangles and Circles: Scalene, Isosceles, Equilateral, and Right Triangles

This review covers essential concepts of triangle types including scalene, isosceles, equilateral, and right triangles. Scalene triangles have no equal sides or angles, while isosceles triangles have two equal sides and corresponding angles. Equilateral triangles feature three equal sides and angles set at 60 degrees each. The right triangle can be scalene or isosceles, emphasizing the importance of side lengths and the Pythagorean theorem. Additionally, we explore central angles within circles, highlighting their properties and calculations.

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Understanding Triangles and Circles: Scalene, Isosceles, Equilateral, and Right Triangles

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  1. GHSGT Review Triangles and Circles

  2. Scalene Triangles • A scalene triangle has NO equal sides and NO equal angles • The longest side is across from the largest angle, and the shortest across from the smallest 70⁰ 10 cm 15 cm 80⁰ 30⁰ 12 cm

  3. Isosceles Triangles • All isosceles triangles have two equal sides • The two angles across from those sides are equal to each other also 30⁰ 7 cm 7 cm 75⁰ 75⁰ 4 cm

  4. Equilateral Triangles • Equilateral means equal-sided: all three sides are equal • Equilateral triangles also have three angles that are all equal (equiangular) • Each angle measures 60⁰ because 180⁰ divided into three equal angles is 60⁰ each. 60⁰ 10 cm 10 cm 10 cm 60⁰ 60⁰

  5. Right Triangles • A right triangle can be either scalene or isosceles, based on the leg lengths • Every right triangle has two legs and a hypotenuse; the hypotenuse is always the longest side, and it’s across from the right angle 45⁰ 45⁰ 67⁰ 33⁰ 5 cm 13 cm 12 cm 8 cm 11.3 cm 8 cm

  6. The Pythagorean Theorem • The Pythagorean Theorem only works for right triangles; it can find the length of a missing side • Use , where a and b can be either leg, but c must be the hypotenuse length (9)² + (12)² = (x)² 81 + 144 = x² 225 = x² 15 cm = x 9 cm x 12 cm

  7. Another Pythagorean Example • Sometimes you need to cancel with the Pythagorean Theorem (10)² + (x)² = (26)² 100 + x² = 576 -100 -100 x² = 476 x = 24 in 10 in 26 in x

  8. Central Angles • When diameters or radii are drawn in a circle, they form central angles • The sum of the central angles of a circle is 360⁰ • A semicircle (half-circle) would be 180⁰ • This angle would measure 123⁰ because 180 + 57 = 237, and 360 – 237 = 123 degrees left over from the circle 180⁰ 57⁰

  9. 1. In this drawing, the length of side A equals 24 inches. The length of side C is 26 inches. Which formula would determine the length of side B? • A. • B. • C. • D. A C 24 in 26 in B

  10. 2. Beth wants to make a design with a circle divided into pie-shaped pieces of equal size. What is the smallest number of pieces Beth can have if she wants the central angles to be right angles? • A. 2 • B. 3 • C. 4 • D. 5

  11. 3. A triangle has side lengths 6, 12, and 12. What type of triangle is it? • A. equilateral • B. isosceles • C. right • D. scalene

  12. Solutions • 1. D – to most easily determine the length of side B, you would subtract those squares and then find the square root • 2. C – Beth’s design would have pieces that are ¼ of the circle • 3. B – Two equal lengths would mean the triangle is isosceles

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