Answer:
7. 12.2 ft
8. 7.07 m
9. 6.06 in
Explanation:
You want the hypotenuse of at right triangle with legs 7 and 10 ft, the diagonal of a square with perimeter 20 m, and the altitude of an equilateral triangle with side length 7 inches.
Pythagorean theorem
These problems all make use of the Pythagorean theorem, which tells yo the relation between sides a, b and hypotenuse c of a right triangle is ...
a² +b² = c²
7. Leash
The length of the leash (c) is ...
c² = (7 ft)² +(10 ft)² = 149 ft²
c = √149 ft ≈ 12.2 ft
The leash is about 12.2 feet long.
8. Square
Each side of the square is 1/4 of the perimeter, so is 5 m. The diagonal is the hypotenuse of a right triangle with legs that length. Its measure is ...
c = √(5² +5²) = 5√2 ≈ 7.07 . . . . meters
The length of the diagonal is about 7.07 meters.
9. Altitude
The altitude of an equilateral triangle is a perpendicular bisector of one side. Its length is ...
b = √(c² -a²)
b = √(7² -3.5²) = √36.75 ≈ 6.06 . . . . inches
The altitude of the triangle is about 6.06 inches.
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Additional comment
The triangles of problems 8 and 9 are known as the two "special" right triangles.
The side lengths of an isosceles right triangle (angles 45°-45°-90°) have the ratios 1 : 1 : √2.
The side lengths of a triangle with angles 30°-60°-90° have the ratios 1 : √3 : 2.
That is, the diagonal of the square is √2 times its side length, and the altitude of the equilateral triangle is (√3)/2 times is side length.
It can be useful to remember these triangles and their side ratios.
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