Properties of Isosceles and Equilateral Triangles

Q.

Can we say that every equilateral triangle is an isosceles triangle?

Q.

In an equilateral triangle of side $2\sqrt{3}cm$, the circumradius is

1. $1$

2. $\sqrt{3}$

3. $2$

4. $2\sqrt{3}$

Q.

ABCD is a square. AC and BD intersect at O. State the measure of $\angle AOB$.

Q.

Question 2 (b)
Draw rough sketches for the following:
In $\Delta$ PQR, PQ, and PR are altitudes of the triangle.

Q. Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles are not congruent.

Q.

Question 17

In $\Delta PQR$, if PQ = QR and $\angle Q={100}^{\circ }$, then $\angle R$ is equal to

a) ${40}^{\circ }$
b) ${80}^{\circ }$
c) ${120}^{\circ }$
d) ${50}^{\circ }$

Q.

Which of the following can be the lengths of the sides of a triangle? Write Yes or No.

$5cm,6cm,11cm$

Q.

If in a triangle, $a^2+b^2+c^2=ca+ab\sqrt{3}$ then the triangle is

1. equilateral

2. isosceles

3. right angled with $A=90°,B=60°andC=30°$

4. None of these

Q.

Question 85

State whether the statement is True or False.

If two angles of a triangle are equal, the third angle is also equal to each of the other two angles.

Q.

The values of $x$ for which the angle between $\overrightarrow{a}=2x^2\hat{i}+4x\hat{j}+\hat{k}$and $\overrightarrow{b}=7\hat{i}-2\hat{j}+x\hat{k}$ is obtuse?

1. $0<x<\frac{1}{2}$

2. $\frac{1}{2}<x<15$

3. $x>\frac{1}{2}$ or $x<0$

4. Form a null set

Q.

In $\mathrm{\Delta }\mathrm{ABC}$, if $\cos \mathrm{A}=\sin \mathrm{B}-\cos \mathrm{C}$ then show that it is a right-angled triangle.

Q. Question 4
ABC is an isosceles triangle right angled at C. Prove that $A{B}^2=2A{C}^2$.

Q.

Question 26

If the exterior angle of a triangle is ${130}^{\circ }$ and its interior opposite angles are equal, then measure of each interior opposite angle is

a) ${55}^{\circ }$
b) ${65}^{\circ }$
c) ${50}^{\circ }$
d) ${60}^{\circ }$

Q. A square with one diagonal has two right angled triangles.

1. True
2. False

Q.

ABC is a triangle in which AB = AC. The base BC is produced to D. If $\angle$ACD = ${130}^{\circ }$, then $\angle$A = ___.

1. 60°

2. 50°

3. 40°

4. 80°

Q. Question 18
Prove that the area of the equilateral triangle drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the equilateral triangle drawn on the other two sides of the triangle.

Q.

The sides $BC,CA,$ and $AB$ of a $∆ABC$are of lengths $a$, $b$ and $c$ respectively. If $D$ is the mid-point of $BC$and $AD$ is perpendicular to $AC$, then the value of $\cos \left(A\right)\cos \left(C\right)$ is

1. $\frac{2(c^2–a^2)}{3ac}$

2. $\frac{3(c^2–a^2)}{2ac}$

3. $1$

4. None of these

Q.

For any two sets $A$ and $B$, $A–(A–B)$ equals

1. $B$

2. $A–B$

3. $A\cap B$

4. $A^c\cap B^c$

Q.

Name the type of the triangle.

$∆PQRwith\angle Q=90°$

Q.

In the given figure:

If AB = 5 cm , QR = 7 cm; then find the value of AC (in cm), if the perimeter of
$△ABC$ is 18 cm.

1. 5

2. 8

3. 7

4. 6

Q.

Read the following statement: An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are 60° each. Define the terms used in this definition that you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in an equilateral triangle?

Q.

If $a=i+2j+2k,\left|b\right|=5$ and the angle between $aandbis\frac{\pi }{6}$, then the area of the triangle formed by these two vectors are two sides is:

1. $\frac{15}{4}$

2. $\frac{15}{2}$

3. $15$

4. $\frac{15\sqrt{3}}{2}$

5. $15\sqrt{3}$

Q.

What is the sum of all the exterior angles of a triangle?

1. ${180}^{\circ }$

2. ${360}^{\circ }$

3. ${540}^{\circ }$

4. ${100}^{\circ }$

Q.

In a triangle $ABC$, if $\left|\overrightarrow{AB}\right|=7$, $\left|\overrightarrow{BC}\right|=5$ and $\left|\overrightarrow{CA}\right|=3$, if the projection of $\overrightarrow{BC}$ on $\overrightarrow{CA}$ is equal to $\frac{n}{2}$, then the value of $n$ is

Q. Match the figures with their respective unknown angles.

1. 80${}^{\circ }$
2. 100${}^{\circ }$
3. 40${}^{\circ }$

Q. Question 9

Area of $\Delta MNO$ in the figure is


(a) $\frac{1}{2}MN\times NO$
(b) $\frac{1}{2}NO\times MO$
(c) $\frac{1}{2}MN\times OQ$
(d) $\frac{1}{2}NO\times OQ$

Q.

Using properties of similar triangles, prove that the line segment joining the mid-points of two sides of a triangle is:

(i) parallel to the third side.

(ii) one-half of the third side.

Q.

Prove that any two sides of a triangle are together greater than twice the median drawn to the third side. [4 MARKS]

Q. Explain the three types of triangles based on their angles. [2 MARKS]

Q. If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.