51
Constructing the mean proportional of two given segments
Textbook: pp. 136–137
GoalConstruct the mean proportional of two given segments a and b with compass and ruler; obtain segments of length √(ab), √2, √5.
New words
construction · yasashsemicircle · yarim aylana√(ab) · √(ab)angle subtending a diameter · diametrga tiralgan burchak
Explanation
The mean proportional comes from the altitude: the altitude to the hypotenuse, which it splits into a and b, has length √(ab). Construction: on a line lay off AB = a and BC = b, find the midpoint O of AC and draw the semicircle with diameter AC. Draw the perpendicular to AC at B; it meets the semicircle at D. The angle ADC subtends a diameter, so it is 90°, hence BD = √(ab). With a unit segment, a = 1 and b = 2 gives √2, and a = 1, b = 5 gives a segment of length √5. The compass point is sharp, so handle it carefully.
Worked examples
a = 3, b = 12: BD = √36 = 6.
a = 1, b = 5: AC = 6, radius 3, and BD = √5 — a segment of length √5.
Class activity
“√ with a compass”: lay off 1 cm and 4 cm in your notebook, construct the semicircle and the perpendicular; measure BD with a ruler and check it is 2 cm.
Practice
1
What is the length of the mean proportional of segments 4 and 9?
6
2
What is the mean proportional of 3 and 27?
9
3
In the construction with AB = 2 and BC = 18, what is BD?
6
4
Why is the angle ADC equal to 90° in the construction?
It is an inscribed angle subtending a diameter.