Questions: Similar Right Triangles and the Geometric Mean
4 questions to test your understanding
Score: 0 / 4
Question 1 Multiple Choice
In right triangle ABC with right angle at C, the altitude from C meets hypotenuse AB at D. If AD = 4 and DB = 9, what is CD?
A5
B6
C6.5
D13
The altitude to the hypotenuse is the geometric mean of the two segments: CD² = AD · DB = 4 · 9 = 36, so CD = 6. The value 6.5 is the arithmetic mean and 13 is the whole hypotenuse.
Question 2 True / False
The altitude drawn from the right angle to the hypotenuse always divides the hypotenuse into two equal parts.
TTrue
FFalse
Answer: False
In the triangle with segments 4 and 9 the parts are unequal. The segments are equal only for an isosceles right triangle, where the altitude is also a median.
Question 3 Short Answer
Using the same triangle (AD = 4, DB = 9), find the length of leg AC.
Think about your answer, then reveal below.
Model answer: AC = √(AD · AB) = √(4 · 13) = √52 = 2√13
Leg AC is the geometric mean of the whole hypotenuse AB = 13 and the segment AD adjacent to it. A check: BC = √(9 · 13) = √117, and AC² + BC² = 52 + 117 = 169 = 13².
Question 4 Short Answer
Why are triangles ACD and ABC similar?
Think about your answer, then reveal below.
Model answer: Both have a right angle (at D and at C) and they share angle A, so they are similar by AA.
Two pairs of congruent angles are enough for similarity. The same reasoning with shared angle B shows triangle CBD is similar to triangle ABC, so all three triangles are similar to one another.