# G.8 and *G.8a - Pythagorean Theorem

G.8    The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry.

G.8  The student will solve problems, including practical problems, involving right triangles.  This will include applying

a)  the Pythagorean Theorem and its converse;

Bloom's Level:  Analyze

### BIG IDEAS

• I can determine what size TV to purchase, figure what length ladder will be needed for a job, and find the shortest route to travel.  I can find the distance across a lake without swimming across it.
• I will calculate the length of a segment and determine a right angle without directly measuring.

### UNDERSTANDING THE STANDARD

• The Pythagorean Theorem is essential for solving problems involving right triangles.
• Many historical and algebraic proofs of the Pythagorean Theorem exist.
• The relationships between the sides and angles of right triangles are useful in many applied fields.
• Some practical problems can be solved by choosing an efficient representation of the problem.
• Another formula for the area of a triangle is A = 1/2 ab sin C.
• The ratios of side lengths in similar right triangles (adjacent/hypotenuse or opposite/hypotenuse) are independent of the scale factor and depend only on the angle the hypotenuse makes with the adjacent side, thus justifying the definition and calculation of trigonometric functions using the ratios of side lengths for similar right triangles.

·  The converse of the Pythagorean Theorem can be used to determine if a triangle is a right triangle.

·  45°-45°-90° and 30°-60°-90° triangles are special right triangles because their side lengths can be specified as exact values using radicals rather than decimal approximations.

·  The sine of an acute angle in a right triangle is equal to the cosine of its complement.

### ESSENTIALS

G.81  Determine whether a triangle formed w/3 given lengths is a right triangle.

·  G.8a1  Determine whether a triangle formed with three given lengths is a right triangle.

·  G.8abc1  Solve problems, including practical problems, using right triangle trigonometry and properties of special right triangles.

### KEY VOCABULARY

triangle, side, angle, right angle, right triangle, Pythagorean Theorem, Converse of Pythagorean Theorem, special right triangle, right triangle trigonometry, length, geometric figure, 45°-45°-90° triangle, 30°-60°-90° triangle, sine, cosine, tangent, grade, hypotenuse, short leg, long leg, complementary angle, angle of elevation, angle of depression, ratio, similar, adjacent, adjacent side, opposite, opposite, scale factor, justify, calculate

Updated: Dec 14, 2017