The Triangle and Its Properties Test

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The Triangle and Its Properties Test - 22

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Question 1
1 / -0

Find the measure of base angles and exterior angle at the vertex of an isosceles triangle if the vertex angle measures $$45^o$$.

A
Base angle $$= 67.5^o$$, exterior angle $$= 135^o$$

B
Base angle $$= 75^o$$, exterior angle $$= 135^o$$

C
Base angle $$= 135^o$$, exterior angle $$= 45^o$$

D
Base angle $$= 67.5^o$$, exterior angle $$= 75^o$$

Solution

Let $$x$$ be the exterior angle at vertex (in degrees).
Then the exterior angle and the interior angle at vertex form a linear pair.
Then,
$$x + 45^o = 180^o$$
$$x = 180^o - 45^o = 135^o$$

The $$2$$ base angles of an isosceles triangle are equal, let us represent each as $$z$$.
Applying angle sum property,
$$z + z + 45 = 180^o$$
$$\Rightarrow 2z = 180^o - 45^o$$
$$\Rightarrow 2z=135^o$$
$$\Rightarrow z=67.5^o$$

So, base angle $$= 67.5^o$$ , exterior angle $$= 135^o$$
Question 2
1 / -0

How many medians(drawn in fig.) are there in a triangle $$PQR$$?

A
1

B
2

C
3

D
4

Solution

A median of a triangle is the line segment that joins any vertex of the triangle with the mid-point of its opposite side.
So, here the triangle is having two medians.
Question 3
1 / -0

In $$\triangle PQR$$, the line segment ____ is the median from vertex $$P$$ to the opposite side $$QR$$ at $$O$$.

A
$$\overline{QT}$$

B
$$\overline{PO}$$

C
$$\overline{OR}$$

D
$$\overline{QO}$$

Solution

A median of a triangle is the line segment that joins any vertex of the triangle with the mid-point of its opposite side.
Therefore, $$\overline{PO}$$ is the line segment for the median $$O$$.
Question 4
1 / -0

In a triangle $$ABC$$, if $$AB = 4$$ and $$AC = 3$$, what is the length of $$BC$$?

A
$$4$$

B
$$5$$

C
$$5.2$$

D
$$6$$

Solution

Using Pythagoras theorem, we will find $$BC$$,
$$AB^{2}+AC^{2}=BC^{2}$$
$$4^{2}+3^{2}=BC^{2}$$
$$16+9=BC^{2}$$
$$BC = 5$$
Question 5
1 / -0

In $$\triangle ABC$$, if $$\angle B = \angle C = 45^{\circ}$$, which of the following is correct?

A
$$\angle A= 60°$$

B
$$\angle A= 70°$$

C
$$\angle A= 80°$$

D
$$\angle A= 90°$$

Solution

As $$\angle B=\angle C=45^{\circ}$$ and by angle sum property, the sum of angles is $$180^o$$.
$$\therefore \angle A+\angle B+\angle C=180^{\circ}$$
$$\Rightarrow \boxed{\angle A=90^{\circ}}$$
Question 6
1 / -0

The distance from town A to town B is five miles. C is six miles from B. Which of the following could be the distance from A to C?
I. $$11$$
II. $$1$$
III. $$7$$

A
I only

B
II only

C
I and II only

D
II and III only

E
I, II or III

Solution

The distance from two A nad B is five miles. C is six miles from B To find distance between A to C there are three possibilities.
Case: $$1$$ In triangle $$ABC,AB=5$$ and $$BC=6$$
Here , the length of AC must be less than the sum of other two sides and greater than the difference of the other sides
$$6.5<6+5$$
$$\Rightarrow 1<Ac<11$$
Thus $$AC=7$$ is a possible
$$\therefore $$ distance from $$A$$ to $$C$$ could be $$11,1 $$ or $$7$$ miles.
Question 7
1 / -0

If each side of an equilateral triangle is doubled then its angle will ______

A
become half

B
be doubled

C
be tripled

D
remain same

Solution

If each side of an equilateral triangle is doubled then its angle will remain same.As if each side is doubled again an equilateral triangle will be formed. So angle remain the same.
Hence option $$D$$ is correct.
Question 8
1 / -0

A line is drawn through the diagonal of a rectangle as shown above. What is the length of the diagonal?

A
5

B
12

C
35

D
37

Solution
Question 9
1 / -0

Find the length of the side labeled $$x$$. The triangle represented in the figure is a right triangle.

A
$$18$$

B
$$24$$

C
$$20$$

D
$$25$$

E
$$22$$

Solution

As it is given that the triangle shown in the figure is a right angle triangle so, by using Pythagoras theorem. We have,
$$\begin{aligned}{}{x^2} + {15^2} &= {25^2}\\{x^2} + 225 &= 625\\x^2& = 625 - 225\\& = 400\\x& = 20\end{aligned}$$

Hence, option $$C$$ is correct.
Question 10
1 / -0

In a triangle with sides of $$7$$ and $$9$$, the third side must be

A
more than $$16$$

B
between $$7$$ and $$9$$

C
between $$2$$ and $$16$$

D
between $$7$$ and $$16$$

E
between $$9$$ and $$16$$

Solution

Given that two sides of triangle are $$7,9$$
Let the third side be $$x$$
We have $$7+9>x$$ and $$x+9>7$$ and $$x+7>9$$ and $$x>0$$
We get $$x<16$$ and $$x>2$$
Therefore the third side will lie between $$2$$ and $$16$$.