Lesson 5 · from §107 of the 1910 text
Geometrical Problems
Altitudes of triangles and trapezoids
The altitude of a triangle whose three sides are known can be found from a system of two quadratic equations.
Example. The sides of a triangle are 6, 8, and 10. Find the altitude on the side 10.
The altitude y divides the side 10 into two parts, x and 10 - x. By the Pythagorean theorem in the two right triangles,
\begin{cases} x^2 + y^2 = 36 & (1) \\ (10 - x)^2 + y^2 = 64 & (2) \end{cases}
Subtracting (1) from (2), 100 - 20x = 28, so x = 3.6. Then y^2 = 36 - 12.96 = 23.04, and the altitude y is 4.8.
The squares of y cancel when the equations are subtracted, leaving a linear equation in x. That is why this system, though of the second degree, is so easily solved.
The area of the triangle is then \frac12 \times \text{base} \times \text{altitude}.
Trapezoids. Let ABCD be a trapezoid with parallel sides AB = 26 and DC = 14 and nonparallel sides 10 each. Draw CE parallel to DA and CF perpendicular to AB. Then EC = 10, AE = 14, and EB = 26 - 14 = 12. If we let EF = x and FC = y, then FB = 12 - x, and
\begin{cases} x^2 + y^2 = 100 \\ (12 - x)^2 + y^2 = 100 \end{cases}
So the trapezoid problem becomes a triangle problem: triangle EBC has sides 10, 10, and 12. The area of a trapezoid is \frac12(\text{sum of the parallel sides}) \times \text{altitude}.
If x comes out negative, the foot of the altitude falls outside the side; the altitude is still correct.
Quick check
The sides of a triangle are 3, 4, and 5. Find the altitude on the side 5.
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.
The sides of a triangle are 6, 8, and 10. Find the altitude on the side 10.
The sides of a triangle are 5, 12, and 13. Find the altitude on the side 13 and the area of the triangle.
The sides of a triangle are 13, 14, and 15. Find the altitude on the side 14 and the area of the triangle.
The sides of a triangle are 9, 10, and 17. Find the altitude on the side 9 and the area of the triangle.
The sides of a triangle are 11, 13, and 20. Find the altitude on the side 11 and the area of the triangle.
The parallel sides of a trapezoid are 14 and 26 respectively, and the two nonparallel sides are 10 each. Find the altitude of the trapezoid.
The two nonparallel sides of a trapezoid are 6 and 8 respectively, and the two bases are 4 and 14 respectively. Find the altitude of the trapezoid.
The bases of a trapezoid are 10 and 16 respectively, and the two nonparallel sides are 25 and 29. Find the altitude of the trapezoid and the area.
The sides of a trapezoid are 10, 20, 17, and 41. The sides 20 and 41 are the bases. Find the altitude and the area.
The sides of a trapezoid are 12, 25, 24, and 17. The sides 12 and 24 are parallel. Find the altitude and the area of the trapezoid.
The sides of a trapezoid are 9, 29, 30, and x. The sides 9 and 30 are the bases, and each is perpendicular to x. Find x and the area of the trapezoid.
The parallel sides of a trapezoid are 15 and 80; the other sides are 72 and 97. Find the altitude and the area of the trapezoid.