Question

In which of the following cases, a unique triangle can be drawn .

• A. $\mathrm{AB}=4\mathrm{cm},\mathrm{BC}=8\mathrm{cm},\mathrm{CA}=2\mathrm{cm}$
• B. $\mathrm{BC}=5.2\mathrm{cm},\angle \mathrm{B}=90°,\angle \mathrm{C}=110°$
• C. $\mathrm{XY}=5\mathrm{cm},\angle \mathrm{X}=45°,\angle \mathrm{Y}=60°$
• D. An isosceles triangle with the length of each equal side $6.2\mathrm{cm}$.

Answer 1

The correct option is C

$\mathrm{XY}=5\mathrm{cm},\angle \mathrm{X}=45°,\angle \mathrm{Y}=60°$

Explanation of the correct option.

Option $\left(C\right)$: $\mathrm{XY}=5\mathrm{cm},\angle \mathrm{X}=45°,\angle \mathrm{Y}=60°$

We know that a triangle can be drawn if sum of all angles is equal to $180°$.

Since, sum of two angles is less than the $180°$.

Thus from the given data we can construct the triangle.

Hence option $\left(C\right)$ is correct.

Explanation of the incorrect option.

Option $\left(A\right)$: $\mathrm{AB}=4\mathrm{cm},\mathrm{BC}=8\mathrm{cm},\mathrm{CA}=2\mathrm{cm}$

We know that a triangle can be drawn if sum of any two sides is greater than the third side.

Since,

$\begin{array}{rcl}BC& >& AB+CA\\ 8& >& 4+2\\ 8& >& 6\end{array}$

Hence option $\left(A\right)$ is incorrect.

Option $\left(B\right)$: $\mathrm{BC}=5.2\mathrm{cm},\angle \mathrm{B}=90°,\angle \mathrm{C}=110°$

We know that a triangle can be drawn if sum of all angles is equal to $180°$.

Since, sum of only two angles is greater than the $180°$.

$$\begin{gathered} \angle B+\angle C=90°+110° \\ \angle B+\angle C=200° \\ \angle B+\angle C>180° \end{gathered}$$

Hence option $\left(B\right)$ is incorrect.

Option $\left(D\right)$: An isosceles triangle with the length of each equal side $6.2\mathrm{cm}$.

We can draw more than one isosceles triangle with side length 6.2 cm.

So, triangle is not unique.

Hence option $\left(D\right)$ is incorrect.

Hence, option (C) is the correct answer.