We will solve different types of problems on properties of triangle.
1. If in any triangle the angles be to one another as 1 : 2 : 3, prove that the corresponding sides are 1 : √3 : 2.
Solution:
Let the angles be k, 2k and 3k.
Then, k + 2k + 3k = 180°
⇒ 6k = 180°
⇒ k = 30°
So, the angles are 30°, 60° and 90°
Let x, y, and z denote the sides opposite to these angles.
Then, x/sin 30° = y/sin 60° = c/sin 90°
⇒ x : y : z = sin 30° : sin 60° : sin 90°
⇒ x : y : z = ½ : √3/2 : 1
⇒ x : y : z = 1 : √3 : 2.
2. Find the lengths of the sides of a triangle, if its angles are in the ratio 1 : 2 : 3 and the circum-radius is 10 cm.,
Solution:
According to the problem, the angles of the triangle are in the ratio 1 : 2 : 3 hence, we assume that the angles are k, 2k, and 3k
i.e., A = k, B = 2k and C = 3k.
Now, A + B + C= 180°
⇒ k + 2k + 3k = 180°
⇒ 6k = 180°
⇒ k = 30°
Therefore, the angles of the triangle are:
A = k = 30°, B = 2k = 60° and C = 3k = 90°
Again, the circum-radius = R = 10 cm.
Therefore, if the lengths of the sides of the triangle be a, b,c then
A = 2R sin A = 2 ∙ 10 ∙ sin 30° = 10 cm.;
B = 2R sin B= 2 ∙ 10 ∙ sin 60° = 10√3 cm.; and
C = 2R sin C = 2 ∙ 10 ∙ sin 90° = 20 cm.
3. If a : b : c = 2 : 3 : 4 and s = 27 inches, find the area of the triangle ABC.
Solution:
Since, a : b : c = 2 : 3 : 4
Let us assume, a = 2x, b = 3x and c = 4x.
Therefore, a + b + c = 2x + 3x + 4x = 9x
Therefore, 9x = 2s
⇒ 9x = 2 × 27, [Since, a + b + c = 2s]
⇒ x = 6
Therefore, the lengths of the three sides are 2 × 6 inches, 3 × 6 inches and 4 × 6 inches i.e., 12 inches, 18 inches and 24 inches.
Therefore, the area of the triangle ABC
= √(s(s - a)(s - b) (s - c))
= √(27.(27 - 12)(27 - 18) (27 - 24)) sq. inches.
= √(27 ∙ 15 ∙ 9 ∙ 3) sq. inches.
= 27√15 sq. inches.
11 and 12 Grade Math
From Problems on Properties of Triangle to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.