If the ratio of the angles of a triangle is 3 : 5 : 7, find the value of the largest angle. A. 36° B. 60° C. 84° D. 15°
C
Finding Triangle Angles from a Ratio
The question asks us to find the value of the largest angle in a triangle, given that the ratio of its angles is 3 : 5 : 7.
Understanding the Properties of Triangles
A fundamental property of any triangle is that the sum of its interior angles is always 180 degrees. This property is key to solving this problem.
Representing the Angles
Given the ratio of the angles is 3 : 5 : 7, we can represent the angles as multiples of a common variable. Let the common variable be \(x\). So, the three angles of the triangle can be represented as:
- First angle: \(3x\)
- Second angle: \(5x\)
- Third angle: \(7x\)
Setting up the Equation
Since the sum of the angles in a triangle is 180 degrees, we can write the equation:
\(3x + 5x + 7x = 180^\circ\)
Solving for the Variable \(x\)
Combine the terms on the left side of the equation:
\(15x = 180^\circ\)
Now, solve for \(x\) by dividing both sides by 15:
\(x = \frac{180^\circ}{15}\)
\(x = 12^\circ\)
Calculating the Actual Angle Values
Now that we have the value of \(x\), we can find the measure of each angle:
- First angle: \(3x = 3 \times 12^\circ = 36^\circ\)
- Second angle: \(5x = 5 \times 12^\circ = 60^\circ\)
- Third angle: \(7x = 7 \times 12^\circ = 84^\circ\)
Let's verify if the sum of these angles is 180 degrees:
\(36^\circ + 60^\circ + 84^\circ = 96^\circ + 84^\circ = 180^\circ\)
The sum is indeed 180 degrees, confirming our calculations for the angles are correct.
Identifying the Largest Angle
The three angles are 36°, 60°, and 84°. The largest among these is 84°.
Comparing with Options
Let's compare our calculated largest angle with the given options:
| Option | Value | Match? |
|---|---|---|
| A | 36° | No (This is the smallest angle) |
| B | 60° | No (This is the middle angle) |
| C | 84° | Yes (This is the largest angle) |
| D | 15° | No (This is the value of x) |
The largest angle calculated is 84°, which matches Option C.
Revision Table: Key Triangle Angle Concepts
| Concept | Description |
|---|---|
| Sum of Interior Angles | The sum of the three interior angles of any triangle is always 180°. |
| Angle Ratio | If angles are in a ratio, they can be represented as \(kx\), \(ly\), \(mz\), etc., where \(k:l:m\) is the ratio and \(x\) is a common multiplier. |
| Solving Ratio Problems | Set up an equation where the sum of the terms with \(x\) equals the total value (in this case, 180°), then solve for \(x\). |
Additional Information on Triangle Angles
Triangles are classified based on their angles:
- Acute-angled triangle: All three angles are less than 90°. In our problem, 36°, 60°, and 84° are all less than 90°, so this triangle is acute-angled.
- Right-angled triangle: One angle is exactly 90°. The other two angles must be acute and sum up to 90°.
- Obtuse-angled triangle: One angle is greater than 90°. The other two angles must be acute and sum up to less than 90°.
Knowing the ratio of angles allows us to determine the specific type of triangle based on its angles.
Important Questions from Triangles, Congruence and Similarity
The radius of the circumcircle of an equilateral triangle of √3 unit side, is:
If ΔABC and Δ PQR are similar and \(\rm\frac{BC}{QR} = \frac{1}{3}\) , find \(\rm\frac{ar(\Delta PQR)} {ar(\Delta BCA)}\)
If the angles of a triangle are in the ratio of 2 : 5 : 8, then find the value of the smallest angle.
ABCD is a parallelogram. Side BC is produced to E such that BC = CE. Join AE which intersects side CD at P. The area of triangle ABE is:
Find out the odd statement in relation to a triangle.
A. The longest side is opposite to the greatest angle.
B. The exterior angle of a triangle = the sum of interior opposite angles.
C. The sum of 2 sides is greater than the 3rd side.
D. The square of one side = the sum of the squares of the other two sides
