Equilateral and Isosceles Triangle Properties

Q.

In an equilateral triangle, a perpendicular drawn from one of the vertices to the opposite side bisects the side.

1. True

2. False

Q. Question 1
A square is inscribed in an isosceles right triangle have one angle common. Show that the vertex of the square opposite the vertex of the common angle bisects the hypotenuse.

Q.

In the given equilateral triangle ABC, find the value of sin ${30}^{\circ }$ and cos ${30}^{\circ }$, without using tables.

AD is the perpendicular bisector of BC.

1. $\frac{1}{\sqrt{3}}$, $\sqrt{3}$

2. $\frac{1}{2}$, $\frac{\sqrt{3}}{2}$

3. $\frac{1}{\sqrt{2}}$, $\sqrt{2}$

4. $\frac{1}{2}$, $\frac{1}{2}$

Q. AD is an altitude of an isosceles triangle ABC in which AB = AC.
Show that,
(i) AD bisects BC
(ii) AD bisects ∠A.

Q. In an equilateral triangle ABC of side 'a', AD is the perpendicular drawn onto side BC. Find $ADcot\theta$ in terms of a.

1. $a$
2. $\frac{a}{2}$
3. $\frac{\sqrt{3}a}{2}$
4. $\frac{3a}{2}$

Q. If r is the radius and R is the circumradius. Then $2R\le r$

1. False
2. True

Q. Question 5
ABC is an isosceles triangle with AB = AC. Drawn AP $\perp$ BC to show that $\angle B=\angle C$.

Q.

In the given figure it is given that AB = CF, EF = BD and $\angle$AFE = $\angle$DBC. Then $△$AFE is congruent to $△$CBD by which criterion?

1. SSS

2. ASA

3. None of these

4. SAS

Q. If $sinA:cosA=4:7$, then what is the value of $\frac{7sinA-3cosA}{7sinA+2cosA}$? [2 MARKS]

Q.

If A = ${60}^o$ and B = ${30}^o$, find the value of
$(sinA.cosB+cosA.sinB{)}^2$ + $(cosA.cosB-sinA.sinB{)}^2$.

Q. $\text{If}\sin A=\frac{1}{2}\text{and}\cos B=\frac{1}{\sqrt{2}}$,
then find the value of (A+B).

Q.

In an equilateral triangle ABC , match the following ratios to the values

$$\begin{array}{cc}Trignometricratios& Values\\ \left(i\right)tan{60}^{\circ }& \left(a\right)\frac{1}{\sqrt{3}}\\ \left(ii\right)cot{30}^{\circ }& \left(b\right)\sqrt{3}\\ \left(iii\right)cosec{30}^{\circ }& \left(c\right)2\\ \left(iv\right)sec{30}^{\circ }& \left(d\right)\frac{2}{\sqrt{3}}\end{array}$$

1. (i) –a , (ii) – b , (iii) – a , (iv) – d

2. (i) –b , (ii) – b , (iii) – c , (iv) – d

3. (i) –a , (ii) – b , (iii) – c , (iv) – d

4. (i) –a , (ii) – b , (iii) – c , (iv) – b

Q.

If $A={15}^{\circ }$, find the value of $4\cos 2A\cdot \sin 4A\cdot \tan 3A-1$.

Q.

If A = ${60}^o$ and B = ${30}^o$, find the value of
$(sinA.cosB+cosA.sinB{)}^2$ + $(cosA.cosB-sinA.sinB{)}^2$.

1. -1

2. 2

3. 1

4. 0

Q. In an equilateral triangle ABC of side 'a', AD is the perpendicular drawn onto side BC. Find $ADcot\theta$ in terms of a.

1. $\frac{\sqrt{3}a}{2}$
2. a
3. $\frac{3a}{2}$
4. $\frac{a}{2}$

Q. In triangle $ABC$, medians $AD$ and $BE$ are drawn.If $AD=4,\angle DAB=\frac{\pi }{6}$ and $\angle ABE=\frac{\pi }{3}$, then the area of the $△ABC$ is

1. $\frac{8}{3}$
2. $\frac{16}{3}$
3. $\frac{32}{3\sqrt{3}}$
4. $\frac{64}{3}$

Q. In $\Delta$ ABC, AD is perpendicular to BC, then $\frac{AC}{AB}$ is equal to

1. $\frac{cosB}{cosC}$
2. $\frac{sinC}{sinB}$
3. $\frac{sinB}{sinC}$
4. $\frac{tanB}{tanC}$

Q.

If $A={15}^{\circ }$, find the value of $4\cos 2A\cdot \sin 4A\cdot \tan 3A-1$.

Q. In an equilateral triangle ABC of side a units, AD is the perpendicular drawn onto side BC. Find $ADcot\theta$ in terms of a.