Lesson 3 · from §90 of the 1910 text
Square Roots in Right Triangles
Square roots in geometry
Fact from geometry. In a right triangle with legs a and b and hypotenuse c (the side opposite the right angle), a^2 + b^2 = c^2.
If leg a is 8 and leg b is 15, then 64 + 225 = c^2, so c^2 = 289 and c = \pm 17. Since -17 is not a practical answer, the hypotenuse is 17. To find a leg, subtract: b^2 = c^2 - a^2. The area of a right triangle is half the product of its legs.
In rectangle ABCD the line DB is a diagonal; it is the hypotenuse of the right triangle DAB, so DB^2 = AB^2 + AD^2. The same holds for a square, whose sides are all equal.
Fact from geometry. A line drawn from one vertex of an equilateral triangle to the middle point of the opposite side is perpendicular to it. So in the equilateral triangle ABC, if D is the middle point of AC, then BD is the altitude and BD^2 = AB^2 - AD^2 = AB^2 - \left(\frac{AB}{2}\right)^2.
The area of a triangle is half the product of the base and the altitude.
Example. The side of an equilateral triangle is 8. Then BD^2 = 64 - 16 = 48, so BD = \sqrt{48} = 6.928 to three decimal places, and the area is \tfrac12 \cdot 8 \cdot 6.928 = 27.713.
Typing answers: whole numbers exactly; inexact answers to three decimal places, like
6.928(within about 0.002 is accepted, since a rounded root may be used along the way).
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Right triangles: a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse.
Find the hypotenuse and the area of a right triangle whose legs are 84 and 13.
Find the hypotenuse and the area of a right triangle whose legs are 133 and 156.
Find the hypotenuse and the area of a right triangle whose legs are 645 and 812.
The hypotenuse of a right triangle is 65 and one leg is 56. Find the other leg and the area.
The hypotenuse of a right triangle is 397 and one leg is 325. Find the other leg and the area.
Rectangles and squares
Find the diagonal of a rectangle whose adjacent sides are 24 feet and 143 feet.
One diagonal of a rectangle is 401 and one side is 399. Find the other side and the area.
One diagonal of a rectangle is 677 and one side is 52. Find the perimeter of the rectangle.
A rectangle is 7 yards longer than it is wide. Its perimeter is 102 feet. Find one diagonal.
One diagonal of a square is 74 meters. Find the side, to three decimal places.
The side of a square is 52 inches. Find one diagonal, to three decimal places.
A rectangle is 2.4 times as long as it is wide. One diagonal is 52. Find the length and the width.
The width of a rectangle is 25% less than the length. The diagonal is 100. Find the area.
The length of a rectangle is 10. The diagonal is twice the shorter side. Find the width, to three decimal places.
Equilateral triangles (see the figure in the lesson)
In the equilateral triangle ABC above, BC is 6. Find BD and the area of the triangle, to three decimal places.
If AC is 10, find BD and the area of the triangle, to three decimal places.
If BD is 10, find AB and the area of the triangle, to three decimal places.
The perimeter of an equilateral triangle is 36. Find the altitude, to three decimal places.
The altitude of an equilateral triangle is 25 centimeters. Find one side, to three decimal places.