Lesson 15 · from §95 of the 1910 text
Problems in Simplest Radical Form
Problems in simplest radical form
These problems use the facts of geometry from the lesson on right triangles, but now the answers are exact: leave them in simplest radical form instead of extracting the roots.
- In a right triangle, a^2 + b^2 = c^2, and the area is half the product of the legs.
- The diagonal of a square of side s is \sqrt{2s^2} = s\sqrt2.
- In an equilateral triangle of side s the altitude is \sqrt{s^2 - \left(\tfrac s2\right)^2} = \tfrac{s}{2}\sqrt3, and the area is half the base times the altitude.
Example. The legs of a right triangle are \dfrac{R}{3} and R. The hypotenuse is \sqrt{\dfrac{R^2}{9} + R^2} = \sqrt{\dfrac{10R^2}{9}} = \dfrac{R}{3}\sqrt{10}.
Typing answers:
5sqrt(2),R*sqrt(3)/2(or(R/2)sqrt(3)),R*sqrt(2 - sqrt(3)). The capital letters R and S are typed as capitals.
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.
Obtain answers in simplest radical form.
One leg of a right triangle is 10 and the other is 5. Find the hypotenuse.
The hypotenuse of a right triangle is 10 and one leg is 5. Find the other leg and the area.
The hypotenuse of a right triangle is R and one leg is \dfrac R2. Find the other leg and the area.
Find the diagonal of a square whose side is 10.
Find the sides and the area of a square whose diagonal is 10.
Find the sides and the area of a square whose diagonal is 2R.
The side of an equilateral triangle is 12. Find the altitude and the area.
The side of an equilateral triangle is S. Find the altitude and the area.
The altitude of an equilateral triangle is 10. Find the side and the area.
The legs of a right triangle are equal. Its hypotenuse is 20. Find the legs and the area of the triangle.
The legs of a right triangle are equal and its area is 32. Find the hypotenuse.
The legs of a right triangle are \dfrac R2 and \dfrac{3R}2. Find the hypotenuse.
One leg of a right triangle is \dfrac R5. The hypotenuse is R. Find the other leg.
The legs of a right triangle are R and \dfrac R2\left(\sqrt5 - 1\right). Find the hypotenuse.
The legs of a right triangle are \dfrac R2 and R - \dfrac R2\sqrt3. Find the hypotenuse.
The base of a certain rectangle is \dfrac{2R}{3}\sqrt{4 - \sqrt3} and the altitude is 9R\,\dfrac{\sqrt{4 + \sqrt3}}{10}. Find the area of a second rectangle five times as long and three times as wide as the first.