Construct a right-angled triangle ΔABC with ∠B = 90°, AC = 5 cm. How many different triangles exist with these measurements? [Hint: Note that the other measurements can take any values. Take AC as the base. What values can ∠A and ∠C take so that the other angle is 90°?]
Given: ∠B = 90° and AC = 5 cm (hypotenuse)
Analysis: We only know one angle (90°) and one side (AC = 5 cm). By angle sum property: ∠A + ∠B + ∠C = 180° Since ∠B = 90°: ∠A + ∠C = 90°
Possibilities: ∠A can take any value between 0° and 90° (exclusive) For each ∠A, ∠C = 90° - ∠A Each different value of ∠A gives a different triangle
Examples: If ∠A = 30°, then ∠C = 60° If ∠A = 45°, then ∠C = 45° Each gives a different triangle with the same hypotenuse AC = 5 cm
Answer: Infinitely many different triangles can be formed.
Explanation
The chapter discusses right-angled triangles where one angle is 90°. The hint guides students to recognize that with only one side and one angle specified, the triangle is not uniquely determined. The angle sum property requires ∠A + ∠C = 90° (since ∠B = 90°). As long as ∠A is between 0° and 90°, we can construct a valid right triangle. Each choice of ∠A gives a different triangle with the same hypotenuse length.
Solution Steps
Given: ∠B = 90° and AC = 5 cm (hypotenuse)
By angle sum property: ∠A + ∠C = 90°
∠A can be any value between 0° and 90°
For each ∠A, ∠C = 90° - ∠A
Each different ∠A gives a different triangle
Therefore, infinitely many triangles possible