Question Video: Using the Law of Sines to Calculate the Number of Triangles that can be Formed Mathematics

For a triangle 𝐴𝐵𝐶, 𝑎 = 5 cm, 𝑏 = 9 cm, and 𝑚∠𝐴 = 25°. How many triangles can be formed? [A] An infinite number of triangles [B] Zero triangles [C] One triangle [D] Two triangles [E] Three triangles

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Video Transcript

For a triangle 𝐴𝐵𝐶, 𝑎 is equal to five centimeters, 𝑏 is equal to nine centimeters, and the measure of angle 𝐴 is 25 degrees. How many triangles can be formed? Is it (A) an infinite number of triangles, (B) zero triangles, (C) one triangle, (D) two triangles, or (E) three triangles?

We are told in the question that the measure of angle 𝐴 in the triangle 𝐴𝐵𝐶 is 25 degrees. Since this angle is less than 90 degrees, it is an acute angle. When angle 𝐴 is acute, we know that there are three possibilities in terms of the number of triangles that can be formed. Firstly, when side length 𝑎 is less than the height ℎ of the triangle, then zero triangles can be formed. Secondly, if side length 𝑎 is equal to ℎ or side length 𝑎 is greater than ℎ and side length 𝑎 is greater than side length 𝑏, then one triangle can be formed. Finally, if the height of the triangle ℎ is less than side length 𝑎, which is less than side length 𝑏, then two triangles can be formed.

As we are given the values of side length 𝑎 and side length 𝑏, this immediately rules out option (A) and option (E). When given two side lengths and one angle in a triangle, it is impossible to draw three triangles or an infinite number of triangles. We are told that side length 𝑎 is equal to five centimeters and side length 𝑏 is equal to nine centimeters. Therefore, 𝑎 is less than 𝑏.

Let’s now consider a possible sketch of the triangle from the information given. We can calculate the height ℎ of this potential triangle using our knowledge of right angle trigonometry. The sine ratio tells us that sin 𝜃 is equal to the opposite over the hypotenuse. From our triangle, the sin of 25 degrees is equal to the height ℎ over nine. Multiplying through by nine, we have ℎ is equal to nine multiplied by sin of 25 degrees. Ensuring that our calculator is in degree mode, typing this in gives us ℎ is equal to 3.803 and so on. To one decimal place, the height of the triangle is therefore equal to 3.8 centimeters. This is less than the five-centimeter length of side 𝑎.

We can therefore see that, from the measurements given, ℎ is less than 𝑎, which is less than 𝑏. And this means that two triangles can be formed, where the second possible triangle is as shown. The correct answer is option (D). From the information given, there are two possible triangles that can be formed: one where angle 𝐵 is an acute angle and one where angle 𝐵 is an obtuse angle. Whilst it is not required in this question, we could use the law of sines to calculate the measures of these angles.