Question
Prove that in a triangle, other than an equilateral triangle, angle opposite the longest side is greater than $\frac{2}{3}$ of a right angle.
✓
Answer
Given: In $\triangle\text{ABC, BC}$ is the longest side.
To prove: $\angle\text{BAC}>\frac{2}{3}$ of a right angle, i.e., $\angle\text{BAC}>60^{\circ}$
Construct: Mark a point D on side AC such that AD = AB = BD.
Proof:
In $\triangle\text{ABD,}$
$\because\text{AD = AB = BD}$ (By construction)
$\therefore\angle1=\angle3=\angle4=60^{\circ}$
Now,
$\angle\text{BAC}=\angle1+\angle2=60^{\circ}+\angle2$
but $60^{\circ}=\frac{2}{3}$ of a right angle
So, $\angle\text{BAC}=\frac{2}{3}$ of a right angle + $\angle2$
Hence, $\angle\text{BAC}>\frac{2}{3}$ of a right angle.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
In Fig. the sides BC, CA and AB of a triangle ABC have been produced to D, E and F respectively. If $\angle\text{ACD}=105^\circ$ and $\angle\text{EAF}=45^\circ,$ find all the angles of the triangle ABC.
→The population of Delhi State in different census years is as given below:
Represent the above information with the help of a bar graph.
| Census year | 1961 | 1971 | 1981 | 1991 | 2001 |
| Population in Lakhs | 30 | 55 | 70 | 110 | 150 |
In the given figure, compute the value of x.
AB and CD are respectively the smallest and largest sides of a quadrilateral ABCD. Show that $\angle\text{A}>\angle\text{C}$ and $\angle\text{B}>\angle\text{D}.$
→Two equal circles intersect in P and Q. A straight line through P meets the circles in A and B. Prove that QA = QB.
Draw the graph of the equation $\frac{\text{x}}{3}+\frac{\text{y}}{4}=1.$ Also, find the area of the triangle formed by the line and the coordinates axes.
→Read the following bar graph and answer the following questions:
- What information is given by the bar graph?
- Which state is the largest producer of rice?
- Which state is the largest producer of wheat?
- Which state has total production of rice and wheat at its maximum?
- Which state has total production of wheat and rice at its minimum?
How many cubic centimetres of iron are there in an open box whose external dimension are 36cm, 25cm and 16.5cm, the iron being 1.5cm thick throughout? If 1cm3 of iron weighs 15g, find the weight of the empty box in kilograms.
The total surface area of a cylinder is 462cm2. Its curved surface area is one-third of its total surface area. Find the volume of the cylinder.
Draw the graph of the line 4x + 3y = 24.
- Write the coordinates of the point where this line interects the x-axis and the y-axis.
- Use this graph to find the area of the triangle formed by the graph line and the coordinate axes.